Image coding method based on transform, and device therefor

ABSTRACT

An image decoding method according to the present document comprises the steps of: when an ISP is applied to a current block and the width or height of a sub-partition block is less than or equal to 2, applying LFNST to the transform coefficients to derive modified transform coefficients; deriving residual samples for the objective block on the basis of an inverse primary transform for the modified transform coefficients; and generating a reconstructed picture on the basis of the residual samples for the objective block, wherein when the sub-partition block is an N×2 block or a 2×N block, (M×2) or (2×M) number of modified transform coefficients are derived by applying an LFNST matrix to a predetermined number of transform coefficients in an upper left M×2 area or 2×M (M≤N) area of the sub-partition block.

BACKGROUND OF THE DISCLOSURE Field of the Disclosure

The present disclosure relates to an image coding technique and, moreparticularly, to a method and an apparatus for coding an image based ontransform in an image coding system.

Related Art

Nowadays, the demand for high-resolution and high-quality images/videossuch as 4K, 8K or more ultra high definition (UHD) images/videos hasbeen increasing in various fields. As the image/video data becomeshigher resolution and higher quality, the transmitted information amountor bit amount increases as compared to the conventional image data.Therefore, when image data is transmitted using a medium such as aconventional wired/wireless broadband line or image/video data is storedusing an existing storage medium, the transmission cost and the storagecost thereof are increased.

Further, nowadays, the interest and demand for immersive media such asvirtual reality (VR), artificial reality (AR) content or hologram, orthe like is increasing, and broadcasting for images/videos having imagefeatures different from those of real images, such as a game image isincreasing.

Accordingly, there is a need for a highly efficient image/videocompression technique for effectively compressing and transmitting orstoring, and reproducing information of high resolution and high qualityimages/videos having various features as described above.

SUMMARY OF THE DISCLOSURE

A technical aspect of the present specification is to provide a methodand apparatus for increasing image coding efficiency.

Another technical aspect of the present specification is to provide amethod and apparatus for increasing efficiency in transform indexcoding.

Yet another technical aspect of the present specification is to providean image coding method and apparatus applying LFNST to a sub-partitionblock having a height and width that are equal to 2 or less.

According to an embodiment of the present disclosure, provided herein isan image decoding method performed by a decoding apparatus. The methodmay include the steps of, when ISP is applied to the current block, andwhen a width or height of a sub-partition block is equal to 2 or less,deriving modified transform coefficients that are modified by applyingLFNST to the transform coefficients, deriving residual samples for thecurrent block based on inverse primary transform on the modifiedtransform coefficients, and generating a reconstructed picture based onresidual samples for the current block, wherein, when the sub-partitionblock is an N×2 block or a 2×N block, (M×2) or (2×M) number of modifiedtransform coefficients may be derived by applying the LFNST matrix to apredetermined number of transform coefficients in a top-left M×2 regionor 2×M region (M≤N) of the sub-partition block.

According to another embodiment of the present disclosure, providedherein is an image encoding method performed by an encoding apparatus.The method may include the steps of deriving transform coefficients byapplying a primary transform of the residual samples, when ISP isapplied to the current block, and when a width or height of asub-partition block is equal to 2 or less, deriving modified transformcoefficients that are modified by applying LFNST to the transformcoefficients, and encoding residual information including information onthe modified transform coefficients, wherein, when the sub-partitionblock is an N×2 block or a 2×N block, 8 or 16 modified transformcoefficients may be derived by applying the LFNST matrix to transformcoefficients in a top-left M×2 region or 2×M region (M≤N) of thesub-partition block.

According to yet another embodiment of the present disclosure, providedherein is a digital storage medium that stores image data includingencoded image information and a bitstream generated according to animage encoding method performed by an encoding apparatus.

According to yet another embodiment of the present disclosure, providedherein is a digital storage medium that stores image data includingencoded image information and a bitstream allowing a decoding apparatusto perform the image decoding method.

According to the present specification, it is possible to increaseoverall image/video compression efficiency.

According to the present specification, it is possible to increaseefficiency in transform index coding.

According to the present specification, it is possible to provide animage coding method and apparatus applying LFNST to a sub-partitionblock having a height and width that are equal to 2 or less.

Effects that can be obtained through specific examples of the presentspecification are not limited to the effects listed above. For example,various technical effects that a person having ordinary skill in therelated art can understand or derive from the present specification mayexist. Accordingly, specific effects of the present specification arenot limited to those explicitly described in the present specification,and can include various effects that can be understood or derived fromthe technical characteristics of the present specification.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 schematically illustrates an example of a video/image codingsystem to which the present disclosure is applicable.

FIG. 2 is a diagram schematically illustrating a configuration of avideo/image encoding apparatus to which the present disclosure isapplicable.

FIG. 3 is a diagram schematically illustrating a configuration of avideo/image decoding apparatus to which the present disclosure isapplicable.

FIG. 4 is a structural diagram of a content streaming system to whichthe present specification is applied.

FIG. 5 schematically illustrates a multiple transform techniqueaccording to an embodiment of the present specification.

FIG. 6 shows exemplary intra directional modes of 65 predictiondirections.

FIG. 7 is a diagram illustrating an RST according to an embodiment ofthe present specification.

FIG. 8 is a diagram illustrating a sequence of arranging output data ofa forward primary transform into a one-dimensional vector according toan example.

FIG. 9 is a diagram illustrating a sequence of arranging output data ofa forward secondary transform into a two-dimensional block according toan example.

FIG. 10 is a diagram illustrating wide-angle intra prediction modesaccording to an embodiment of the present specification.

FIG. 11 is a diagram illustrating a block shape to which the LFNST isapplied.

FIG. 12 is a diagram illustrating an arrangement of output data of aforward LFNST according to an example.

FIG. 13 shows that a number of output data for the forward LFNST islimited to a maximum of 16 according to an example.

FIG. 14 is a diagram illustrating a zero-out in a block to which the 4×4LFNST is applied according to an example.

FIG. 15 is a diagram illustrating a zero-out in a block to which the 8×8LFNST is applied according to an example.

FIG. 16 is a diagram illustrating a zero-out in a block to which the 8×8LFNST is applied according to another example.

FIG. 17 illustrates an example of sub-blocks into which a coding blockis divided.

FIG. 18 illustrates another example of sub-blocks into which a codingblock is divided.

FIG. 19 is a diagram illustrating symmetry between M×2 (M×1) blocks and2×M (1×M) blocks according to an example.

FIG. 20 is a diagram illustrating an example of transposing a 2×M blockaccording to an example.

FIG. 21 illustrates a scanning order for 8×2 or 2×8 regions according toan example.

FIG. 22 is a flowchart illustrating an operation of a video decodingapparatus according to an embodiment of the present specification.

FIG. 23 is a flowchart illustrating an operation of a video encodingapparatus according to an embodiment of the present specification.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

While the present disclosure may be susceptible to various modificationsand include various embodiments, specific embodiments thereof have beenshown in the drawings by way of example and will now be described indetail. However, this is not intended to limit the present disclosure tothe specific embodiments disclosed herein. The terminology used hereinis for the purpose of describing specific embodiments only, and is notintended to limit technical idea of the present disclosure. The singularforms may include the plural forms unless the context clearly indicatesotherwise. The terms such as “include” and “have” are intended toindicate that features, numbers, steps, operations, elements,components, or combinations thereof used in the following descriptionexist, and thus should not be understood as that the possibility ofexistence or addition of one or more different features, numbers, steps,operations, elements, components, or combinations thereof is excluded inadvance.

Meanwhile, each component on the drawings described herein isillustrated independently for convenience of description as tocharacteristic functions different from each other, and however, it isnot meant that each component is realized by a separate hardware orsoftware. For example, any two or more of these components may becombined to form a single component, and any single component may bedivided into plural components. The embodiments in which components arecombined and/or divided will belong to the scope of the patent right ofthe present disclosure as long as they do not depart from the essence ofthe present disclosure.

Hereinafter, preferred embodiments of the present disclosure will beexplained in more detail while referring to the attached drawings. Inaddition, the same reference signs are used for the same components onthe drawings, and repeated descriptions for the same components will beomitted.

This document relates to video/image coding. For example, themethod/example disclosed in this document may relate to a VVC (VersatileVideo Coding) standard (ITU-T Rec. H.266), a next-generation video/imagecoding standard after VVC, or other video coding related standards(e.g., HEVC (High Efficiency Video Coding) standard (ITU-T Rec. H.265),EVC (essential video coding) standard, AVS2 standard, etc.).

In this document, a variety of embodiments relating to video/imagecoding may be provided, and, unless specified to the contrary, theembodiments may be combined to each other and be performed.

In this document, a video may mean a set of a series of images overtime. Generally, a picture means a unit representing an image at aspecific time zone, and a slice/tile is a unit constituting a part ofthe picture. The slice/tile may include one or more coding tree units(CTUs). One picture may be constituted by one or more slices/tiles. Onepicture may be constituted by one or more tile groups. One tile groupmay include one or more tiles.

A pixel or a pel may mean a smallest unit constituting one picture (orimage). Also, ‘sample’ may be used as a term corresponding to a pixel. Asample may generally represent a pixel or a value of a pixel, and mayrepresent only a pixel/pixel value of a luma component or only apixel/pixel value of a chroma component. Alternatively, the sample mayrefer to a pixel value in the spatial domain, or when this pixel valueis converted to the frequency domain, it may refer to a transformcoefficient in the frequency domain.

A unit may represent the basic unit of image processing. The unit mayinclude at least one of a specific region and information related to theregion. One unit may include one luma block and two chroma (e.g., cb,cr) blocks. The unit and a term such as a block, an area, or the likemay be used in place of each other according to circumstances. In ageneral case, an M×N block may include a set (or an array) of samples(or sample arrays) or transform coefficients consisting of M columns andN rows.

In this document, the term “/” and “,” should be interpreted to indicate“and/or.” For instance, the expression “A/B” may mean “A and/or B.”Further, “A, B” may mean “A and/or B.” Further, “A/B/C” may mean “atleast one of A, B, and/or C.” Also, “A/B/C” may mean “at least one of A,B, and/or C.”

Further, in the document, the term “or” should be interpreted toindicate “and/or.” For instance, the expression “A or B” may include 1)only A, 2) only B, and/or 3) both A and B. In other words, the term “or”in this document should be interpreted to indicate “additionally oralternatively.”

In the present disclosure, “at least one of A and B” may mean “only A”,“only B”, or “both A and B”. In addition, in the present disclosure, theexpression “at least one of A or B” or “at least one of A and/or B” maybe interpreted as “at least one of A and B”.

In addition, in the present disclosure, “at least one of A, B, and C”may mean “only A”, “only B”, “only C”, or “any combination of A, B, andC”. In addition, “at least one of A, B, or C” or “at least one of A, B,and/or C” may mean “at least one of A, B, and C”.

In addition, a parenthesis used in the present disclosure may mean “forexample”. Specifically, when indicated as “prediction (intraprediction)”, it may mean that “intra prediction” is proposed as anexample of “prediction”. In other words, the “prediction” of the presentdisclosure is not limited to “intra prediction”, and “intra prediction”may be proposed as an example of “prediction”. In addition, whenindicated as “prediction (i.e., intra prediction)”, it may also meanthat “intra prediction” is proposed as an example of “prediction”.

Technical features individually described in one figure in the presentdisclosure may be individually implemented or may be simultaneouslyimplemented.

FIG. 1 schematically illustrates an example of a video/image codingsystem to which the present disclosure is applicable.

Referring to FIG. 1, the video/image coding system may include a firstdevice (source device) and a second device (receive device). The sourcedevice may deliver encoded video/image information or data in the formof a file or streaming to the receive device via a digital storagemedium or network.

The source device may include a video source, an encoding apparatus, anda transmitter. The receive device may include a receiver, a decodingapparatus, and a renderer. The encoding apparatus may be called avideo/image encoding apparatus, and the decoding apparatus may be calleda video/image decoding apparatus. The transmitter may be included in theencoding apparatus. The receiver may be included in the decodingapparatus. The renderer may include a display, and the display may beconfigured as a separate device or an external component.

The video source may obtain a video/image through a process ofcapturing, synthesizing, or generating a video/image. The video sourcemay include a video/image capture device and/or a video/image generatingdevice. The video/image capture device may include, for example, one ormore cameras, video/image archives including previously capturedvideo/images, or the like. The video/image generating device mayinclude, for example, a computer, a tablet and a smartphone, and may(electronically) generate a video/image. For example, a virtualvideo/image may be generated through a computer or the like. In thiscase, the video/image capturing process may be replaced by a process ofgenerating related data.

The encoding apparatus may encode an input video/image. The encodingapparatus may perform a series of procedures such as prediction,transform, and quantization for compression and coding efficiency. Theencoded data (encoded video/image information) may be output in the formof a bitstream.

The transmitter may transmit the encoded video/image information or dataoutput in the form of a bitstream to the receiver of the receive devicethrough a digital storage medium or a network in the form of a file orstreaming. The digital storage medium may include various storagemediums such as USB, SD, CD, DVD, Blu-ray, HDD, SSD, and the like. Thetransmitter may include an element for generating a media file through apredetermined file format, and may include an element for transmissionthrough a broadcast/communication network. The receiver mayreceive/extract the bitstream and transmit the received/extractedbitstream to the decoding apparatus.

The decoding apparatus may decode a video/image by performing a seriesof procedures such as dequantization, inverse transform, prediction, andthe like corresponding to the operation of the encoding apparatus.

The renderer may render the decoded video/image. The renderedvideo/image may be displayed through the display.

FIG. 2 is a diagram schematically illustrating a configuration of avideo/image encoding apparatus to which the present disclosure isapplicable. Hereinafter, what is referred to as the video encodingapparatus may include an image encoding apparatus.

Referring to FIG. 2, the encoding apparatus 200 may include an imagepartitioner 210, a predictor 220, a residual processor 230, an entropyencoder 240, an adder 250, a filter 260, and a memory 270. The predictor220 may include an inter predictor 221 and an intra predictor 222. Theresidual processor 230 may include a transformer 232, a quantizer 233, adequantizer 234, an inverse transformer 235. The residual processor 230may further include a subtractor 231. The adder 250 may be called areconstructor or reconstructed block generator. The image partitioner210, the predictor 220, the residual processor 230, the entropy encoder240, the adder 250, and the filter 260, which have been described above,may be constituted by one or more hardware components (e.g., encoderchipsets or processors) according to an embodiment. Further, the memory270 may include a decoded picture buffer (DPB), and may be constitutedby a digital storage medium. The hardware component may further includethe memory 270 as an internal/external component.

The image partitioner 210 may partition an input image (or a picture ora frame) input to the encoding apparatus 200 into one or more processingunits. As one example, the processing unit may be called a coding unit(CU). In this case, starting with a coding tree unit (CTU) or thelargest coding unit (LCU), the coding unit may be recursivelypartitioned according to the Quad-tree binary-tree ternary-tree (QTBTTT)structure. For example, one coding unit may be divided into a pluralityof coding units of a deeper depth based on the quad-tree structure, thebinary-tree structure, and/or the ternary structure. In this case, forexample, the quad-tree structure may be applied first and thebinary-tree structure and/or the ternary structure may be applied later.Alternatively, the binary-tree structure may be applied first. Thecoding procedure according to the present disclosure may be performedbased on the final coding unit which is not further partitioned. In thiscase, the maximum coding unit may be used directly as a final codingunit based on coding efficiency according to the image characteristic.Alternatively, the coding unit may be recursively partitioned intocoding units of a further deeper depth as needed, so that the codingunit of an optimal size may be used as a final coding unit. Here, thecoding procedure may include procedures such as prediction, transform,and reconstruction, which will be described later. As another example,the processing unit may further include a prediction unit (PU) or atransform unit (TU). In this case, the prediction unit and the transformunit may be split or partitioned from the above-described final codingunit. The prediction unit may be a unit of sample prediction, and thetransform unit may be a unit for deriving a transform coefficient and/ora unit for deriving a residual signal from a transform coefficient.

The unit and a term such as a block, an area, or the like may be used inplace of each other according to circumstances. In a general case, anM×N block may represent a set of samples or transform coefficientsconsisting of M columns and N rows. The sample may generally represent apixel or a value of a pixel, and may represent only a pixel/pixel valueof a luma component, or only a pixel/pixel value of a chroma component.The sample may be used as a term corresponding to a pixel or a pel ofone picture (or image).

The subtractor 231 subtracts a prediction signal (predicted block,prediction sample array) output from the predictor 220 from an inputimage signal (original block, original sample array) to generate aresidual signal (residual block, residual sample array), and thegenerated residual signal is transmitted to the transformer 232. Thepredictor 220 may perform prediction on a processing target block(hereinafter, referred to as ‘current block’), and may generate apredicted block including prediction samples for the current block. Thepredictor 220 may determine whether intra prediction or inter predictionis applied on a current block or CU basis. As discussed later in thedescription of each prediction mode, the predictor may generate variousinformation relating to prediction, such as prediction mode information,and transmit the generated information to the entropy encoder 240. Theinformation on the prediction may be encoded in the entropy encoder 240and output in the form of a bitstream.

The intra predictor 222 may predict the current block by referring tosamples in the current picture. The referred samples may be located inthe neighbor of or apart from the current block according to theprediction mode. In the intra prediction, prediction modes may include aplurality of non-directional modes and a plurality of directional modes.The non-directional modes may include, for example, a DC mode and aplanar mode. The directional mode may include, for example, 33directional prediction modes or 65 directional prediction modesaccording to the degree of detail of the prediction direction. However,this is merely an example, and more or less directional prediction modesmay be used depending on a setting. The intra predictor 222 maydetermine the prediction mode applied to the current block by using theprediction mode applied to the neighboring block.

The inter predictor 221 may derive a predicted block for the currentblock based on a reference block (reference sample array) specified by amotion vector on a reference picture. At this time, in order to reducethe amount of motion information transmitted in the inter predictionmode, the motion information may be predicted on a block, subblock, orsample basis based on correlation of motion information between theneighboring block and the current block. The motion information mayinclude a motion vector and a reference picture index. The motioninformation may further include inter prediction direction (L0prediction, L1 prediction, Bi prediction, etc.) information. In the caseof inter prediction, the neighboring block may include a spatialneighboring block existing in the current picture and a temporalneighboring block existing in the reference picture. The referencepicture including the reference block and the reference pictureincluding the temporal neighboring block may be same to each other ordifferent from each other. The temporal neighboring block may be calleda collocated reference block, a collocated CU (colCU), and the like, andthe reference picture including the temporal neighboring block may becalled a collocated picture (colPic). For example, the inter predictor221 may configure a motion information candidate list based onneighboring blocks and generate information indicating which candidateis used to derive a motion vector and/or a reference picture index ofthe current block. Inter prediction may be performed based on variousprediction modes. For example, in the case of a skip mode and a mergemode, the inter predictor 221 may use motion information of theneighboring block as motion information of the current block. In theskip mode, unlike the merge mode, the residual signal may not betransmitted. In the case of the motion information prediction (motionvector prediction, MVP) mode, the motion vector of the neighboring blockmay be used as a motion vector predictor and the motion vector of thecurrent block may be indicated by signaling a motion vector difference.

The predictor 220 may generate a prediction signal based on variousprediction methods. For example, the predictor may apply intraprediction or inter prediction for prediction on one block, and, aswell, may apply intra prediction and inter prediction at the same time.This may be called combined inter and intra prediction (CIIP). Further,the predictor may be based on an intra block copy (IBC) prediction mode,or a palette mode in order to perform prediction on a block. The IBCprediction mode or palette mode may be used for content image/videocoding of a game or the like, such as screen content coding (SCC).Although the IBC basically performs prediction in a current block, itcan be performed similarly to inter prediction in that it derives areference block in a current block. That is, the IBC may use at leastone of inter prediction techniques described in the present disclosure.

The prediction signal generated through the inter predictor 221 and/orthe intra predictor 222 may be used to generate a reconstructed signalor to generate a residual signal. The transformer 232 may generatetransform coefficients by applying a transform technique to the residualsignal. For example, the transform technique may include at least one ofa discrete cosine transform (DCT), a discrete sine transform (DST), aKarhunen-Loève transform (KLT), a graph-based transform (GBT), or aconditionally non-linear transform (CNT). Here, the GBT means transformobtained from a graph when relationship information between pixels isrepresented by the graph. The CNT refers to transform obtained based ona prediction signal generated using all previously reconstructed pixels.In addition, the transform process may be applied to square pixel blockshaving the same size or may be applied to blocks having a variable sizerather than the square one.

The quantizer 233 may quantize the transform coefficients and transmitthem to the entropy encoder 240, and the entropy encoder 240 may encodethe quantized signal (information on the quantized transformcoefficients) and output the encoded signal in a bitstream. Theinformation on the quantized transform coefficients may be referred toas residual information. The quantizer 233 may rearrange block typequantized transform coefficients into a one-dimensional vector formbased on a coefficient scan order, and generate information on thequantized transform coefficients based on the quantized transformcoefficients of the one-dimensional vector form. The entropy encoder 240may perform various encoding methods such as, for example, exponentialGolomb, context-adaptive variable length coding (CAVLC),context-adaptive binary arithmetic coding (CABAC), and the like. Theentropy encoder 240 may encode information necessary for video/imagereconstruction other than quantized transform coefficients (e.g. valuesof syntax elements, etc.) together or separately. Encoded information(e.g., encoded video/image information) may be transmitted or stored ona unit basis of a network abstraction layer (NAL) in the form of abitstream. The video/image information may further include informationon various parameter sets such as an adaptation parameter set (APS), apicture parameter set (PPS), a sequence parameter set (SPS), a videoparameter set (VPS) or the like. Further, the video/image informationmay further include general constraint information. In the presentdisclosure, information and/or syntax elements which aretransmitted/signaled to the decoding apparatus from the encodingapparatus may be included in video/image information. The video/imageinformation may be encoded through the above-described encodingprocedure and included in the bitstream. The bitstream may betransmitted through a network, or stored in a digital storage medium.Here, the network may include a broadcast network, a communicationnetwork and/or the like, and the digital storage medium may includevarious storage media such as USB, SD, CD, DVD, Blu-ray, HDD, SSD, andthe like. A transmitter (not shown) which transmits a signal output fromthe entropy encoder 240 and/or a storage (not shown) which stores it maybe configured as an internal/external element of the encoding apparatus200, or the transmitter may be included in the entropy encoder 240.

Quantized transform coefficients output from the quantizer 233 may beused to generate a prediction signal. For example, by applyingdequantization and inverse transform to quantized transform coefficientsthrough the dequantizer 234 and the inverse transformer 235, theresidual signal (residual block or residual samples) may bereconstructed. The adder 155 adds the reconstructed residual signal to aprediction signal output from the inter predictor 221 or the intrapredictor 222, so that a reconstructed signal (reconstructed picture,reconstructed block, reconstructed sample array) may be generated. Whenthere is no residual for a processing target block as in a case wherethe skip mode is applied, the predicted block may be used as areconstructed block. The adder 250 may be called a reconstructor or areconstructed block generator. The generated reconstructed signal may beused for intra prediction of a next processing target block in thecurrent block, and as described later, may be used for inter predictionof a next picture through filtering.

Meanwhile, in the picture encoding and/or reconstructing process, lumamapping with chroma scaling (LMCS) may be applied.

The filter 260 may improve subjective/objective video quality byapplying the filtering to the reconstructed signal. For example, thefilter 260 may generate a modified reconstructed picture by applyingvarious filtering methods to the reconstructed picture, and may storethe modified reconstructed picture in the memory 270, specifically inthe DPB of the memory 270. The various filtering methods may include,for example, deblocking filtering, sample adaptive offset, an adaptiveloop filter, a bilateral filter or the like. As discussed later in thedescription of each filtering method, the filter 260 may generatevarious information relating to filtering, and transmit the generatedinformation to the entropy encoder 240. The information on the filteringmay be encoded in the entropy encoder 240 and output in the form of abitstream.

The modified reconstructed picture which has been transmitted to thememory 270 may be used as a reference picture in the inter predictor221. Through this, the encoding apparatus can avoid prediction mismatchin the encoding apparatus 100 and a decoding apparatus when the interprediction is applied, and can also improve coding efficiency.

The memory 270 DPB may store the modified reconstructed picture in orderto use it as a reference picture in the inter predictor 221. The memory270 may store motion information of a block in the current picture, fromwhich motion information has been derived (or encoded) and/or motioninformation of blocks in an already reconstructed picture. The storedmotion information may be transmitted to the inter predictor 221 to beutilized as motion information of a neighboring block or motioninformation of a temporal neighboring block. The memory 270 may storereconstructed samples of reconstructed blocks in the current picture,and transmit them to the intra predictor 222.

FIG. 3 is a diagram schematically illustrating a configuration of avideo/image decoding apparatus to which the present disclosure isapplicable.

Referring to FIG. 3, the video decoding apparatus 300 may include anentropy decoder 310, a residual processor 320, a predictor 330, an adder340, a filter 350 and a memory 360. The predictor 330 may include aninter predictor 331 and an intra predictor 332. The residual processor320 may include a dequantizer 321 and an inverse transformer 321. Theentropy decoder 310, the residual processor 320, the predictor 330, theadder 340, and the filter 350, which have been described above, may beconstituted by one or more hardware components (e.g., decoder chipsetsor processors) according to an embodiment. Further, the memory 360 mayinclude a decoded picture buffer (DPB), and may be constituted by adigital storage medium. The hardware component may further include thememory 360 as an internal/external component.

When a bitstream including video/image information is input, thedecoding apparatus 300 may reconstruct an image correspondingly to aprocess by which video/image information has been processed in theencoding apparatus of FIG. 2. For example, the decoding apparatus 300may derive units/blocks based on information relating to block partitionobtained from the bitstream. The decoding apparatus 300 may performdecoding by using a processing unit applied in the encoding apparatus.Therefore, the processing unit of decoding may be, for example, a codingunit, which may be partitioned along the quad-tree structure, thebinary-tree structure, and/or the ternary-tree structure from a codingtree unit or a largest coding unit. One or more transform units may bederived from the coding unit. And, the reconstructed image signaldecoded and output through the decoding apparatus 300 may be reproducedthrough a reproducer.

The decoding apparatus 300 may receive a signal output from the encodingapparatus of FIG. 2 in the form of a bitstream, and the received signalmay be decoded through the entropy decoder 310. For example, the entropydecoder 310 may parse the bitstream to derive information (e.g.,video/image information) required for image reconstruction (or picturereconstruction). The video/image information may further includeinformation on various parameter sets such as an adaptation parameterset (APS), a picture parameter set (PPS), a sequence parameter set(SPS), a video parameter set (VPS) or the like. Further, the video/imageinformation may further include general constraint information. Thedecoding apparatus may decode a picture further based on information onthe parameter set and/or the general constraint information. In thepresent disclosure, signaled/received information and/or syntaxelements, which will be described later, may be decoded through thedecoding procedure and be obtained from the bitstream. For example, theentropy decoder 310 may decode information in the bitstream based on acoding method such as exponential Golomb encoding, CAVLC, CABAC, or thelike, and may output a value of a syntax element necessary for imagereconstruction and quantized values of a transform coefficient regardinga residual. More specifically, a CABAC entropy decoding method mayreceive a bin corresponding to each syntax element in a bitstream,determine a context model using decoding target syntax elementinformation and decoding information of neighboring and decoding targetblocks, or information of symbol/bin decoded in a previous step, predictbin generation probability according to the determined context model andperform arithmetic decoding of the bin to generate a symbolcorresponding to each syntax element value. Here, the CABAC entropydecoding method may update the context model using information of asymbol/bin decoded for a context model of the next symbol/bin afterdetermination of the context model. Information on prediction amonginformation decoded in the entropy decoder 310 may be provided to thepredictor (inter predictor 332 and intra predictor 331), and residualvalues, that is, quantized transform coefficients, on which entropydecoding has been performed in the entropy decoder 310, and associatedparameter information may be input to the residual processor 320. Theresidual processor 320 may derive a residual signal (residual block,residual samples, residual sample array). Further, information onfiltering among information decoded in the entropy decoder 310 may beprovided to the filter 350. Meanwhile, a receiver (not shown) whichreceives a signal output from the encoding apparatus may furtherconstitute the decoding apparatus 300 as an internal/external element,and the receiver may be a component of the entropy decoder 310.Meanwhile, the decoding apparatus according to the present disclosuremay be called a video/image/picture coding apparatus, and the decodingapparatus may be classified into an information decoder(video/image/picture information decoder) and a sample decoder(video/image/picture sample decoder). The information decoder mayinclude the entropy decoder 310, and the sample decoder may include atleast one of the dequantizer 321, the inverse transformer 322, the adder340, the filter 350, the memory 360, the inter predictor 332, and theintra predictor 331.

The dequantizer 321 may output transform coefficients by dequantizingthe quantized transform coefficients. The dequantizer 321 may rearrangethe quantized transform coefficients in the form of a two-dimensionalblock. In this case, the rearrangement may perform rearrangement basedon an order of coefficient scanning which has been performed in theencoding apparatus. The dequantizer 321 may perform dequantization onthe quantized transform coefficients using quantization parameter (e.g.,quantization step size information), and obtain transform coefficients.

The dequantizer 322 obtains a residual signal (residual block, residualsample array) by inverse transforming transform coefficients.

The predictor may perform prediction on the current block, and generatea predicted block including prediction samples for the current block.The predictor may determine whether intra prediction or inter predictionis applied to the current block based on the information on predictionoutput from the entropy decoder 310, and specifically may determine anintra/inter prediction mode.

The predictor may generate a prediction signal based on variousprediction methods. For example, the predictor may apply intraprediction or inter prediction for prediction on one block, and, aswell, may apply intra prediction and inter prediction at the same time.This may be called combined inter and intra prediction (CIIP). Inaddition, the predictor may perform intra block copy (IBC) forprediction on a block. The intra block copy may be used for contentimage/video coding of a game or the like, such as screen content coding(SCC). Although the IBC basically performs prediction in a currentblock, it can be performed similarly to inter prediction in that itderives a reference block in a current block. That is, the IBC may useat least one of inter prediction techniques described in the presentdisclosure.

The intra predictor 331 may predict the current block by referring tothe samples in the current picture. The referred samples may be locatedin the neighbor of or apart from the current block according to theprediction mode. In the intra prediction, prediction modes may include aplurality of non-directional modes and a plurality of directional modes.The intra predictor 331 may determine the prediction mode applied to thecurrent block by using the prediction mode applied to the neighboringblock.

The inter predictor 332 may derive a predicted block for the currentblock based on a reference block (reference sample array) specified by amotion vector on a reference picture. At this time, in order to reducethe amount of motion information transmitted in the inter predictionmode, the motion information may be predicted on a block, subblock, orsample basis based on correlation of motion information between theneighboring block and the current block. The motion information mayinclude a motion vector and a reference picture index. The motioninformation may further include inter prediction direction (L0prediction, L1 prediction, Bi prediction, etc.) information. In the caseof inter prediction, the neighboring block may include a spatialneighboring block existing in the current picture and a temporalneighboring block existing in the reference picture. For example, theinter predictor 332 may configure a motion information candidate listbased on neighboring blocks, and derive a motion vector and/or areference picture index of the current block based on received candidateselection information. Inter prediction may be performed based onvarious prediction modes, and the information on prediction may includeinformation indicating a mode of inter prediction for the current block.

The adder 340 may generate a reconstructed signal (reconstructedpicture, reconstructed block, reconstructed sample array) by adding theobtained residual signal to the prediction signal (predicted block,prediction sample array) output from the predictor 330. When there is noresidual for a processing target block as in a case where the skip modeis applied, the predicted block may be used as a reconstructed block.

The adder 340 may be called a reconstructor or a reconstructed blockgenerator. The generated reconstructed signal may be used for intraprediction of a next processing target block in the current block, andas described later, may be output through filtering or be used for interprediction of a next picture.

Meanwhile, in the picture decoding process, luma mapping with chromascaling (LMCS) may be applied.

The filter 350 may improve subjective/objective video quality byapplying the filtering to the reconstructed signal. For example, thefilter 350 may generate a modified reconstructed picture by applyingvarious filtering methods to the reconstructed picture, and may transmitthe modified reconstructed picture in the memory 360, specifically inthe DPB of the memory 360. The various filtering methods may include,for example, deblocking filtering, sample adaptive offset, an adaptiveloop filter, a bilateral filter or the like.

The (modified) reconstructed picture which has been stored in the DPB ofthe memory 360 may be used as a reference picture in the inter predictor332. The memory 360 may store motion information of a block in thecurrent picture, from which motion information has been derived (ordecoded) and/or motion information of blocks in an already reconstructedpicture. The stored motion information may be transmitted to the interpredictor 260 to be utilized as motion information of a neighboringblock or motion information of a temporal neighboring block. The memory360 may store reconstructed samples of reconstructed blocks in thecurrent picture, and transmit them to the intra predictor 331.

In this specification, the examples described in the predictor 330, thedequantizer 321, the inverse transformer 322, and the filter 350 of thedecoding apparatus 300 may be similarly or correspondingly applied tothe predictor 220, the dequantizer 234, the inverse transformer 235, andthe filter 260 of the encoding apparatus 200, respectively.

As described above, prediction is performed in order to increasecompression efficiency in performing video coding. Through this, apredicted block including prediction samples for a current block, whichis a coding target block, may be generated. Here, the predicted blockincludes prediction samples in a space domain (or pixel domain). Thepredicted block may be identically derived in the encoding apparatus andthe decoding apparatus, and the encoding apparatus may increase imagecoding efficiency by signaling to the decoding apparatus not originalsample value of an original block itself but information on residual(residual information) between the original block and the predictedblock. The decoding apparatus may derive a residual block includingresidual samples based on the residual information, generate areconstructed block including reconstructed samples by adding theresidual block to the predicted block, and generate a reconstructedpicture including reconstructed blocks.

The residual information may be generated through transform andquantization procedures. For example, the encoding apparatus may derivea residual block between the original block and the predicted block,derive transform coefficients by performing a transform procedure onresidual samples (residual sample array) included in the residual block,and derive quantized transform coefficients by performing a quantizationprocedure on the transform coefficients, so that it may signalassociated residual information to the decoding apparatus (through abitstream). Here, the residual information may include valueinformation, position information, a transform technique, transformkernel, a quantization parameter or the like of the quantized transformcoefficients. The decoding apparatus may perform aquantization/dequantization procedure and derive the residual samples(or residual sample block), based on residual information. The decodingapparatus may generate a reconstructed block based on a predicted blockand the residual block. The encoding apparatus may derive a residualblock by dequantizing/inverse transforming quantized transformcoefficients for reference for inter prediction of a next picture, andmay generate a reconstructed picture based on this.

FIG. 4 illustrates the structure of a content streaming system to whichthe present disclosure is applied.

Further, the contents streaming system to which the present disclosureis applied may largely include an encoding server, a streaming server, aweb server, a media storage, a user equipment, and a multimedia inputdevice.

The encoding server functions to compress to digital data the contentsinput from the multimedia input devices, such as the smart phone, thecamera, the camcoder and the like, to generate a bitstream, and totransmit it to the streaming server. As another example, in a case wherethe multimedia input device, such as, the smart phone, the camera, thecamcoder or the like, directly generates a bitstream, the encodingserver may be omitted. The bitstream may be generated by an encodingmethod or a bitstream generation method to which the present disclosureis applied. And the streaming server may store the bitstream temporarilyduring a process to transmit or receive the bitstream.

The streaming server transmits multimedia data to the user equipment onthe basis of a user's request through the web server, which functions asan instrument that informs a user of what service there is. When theuser requests a service which the user wants, the web server transfersthe request to the streaming server, and the streaming server transmitsmultimedia data to the user. In this regard, the contents streamingsystem may include a separate control server, and in this case, thecontrol server functions to control commands/responses betweenrespective equipments in the content streaming system.

The streaming server may receive contents from the media storage and/orthe encoding server. For example, in a case the contents are receivedfrom the encoding server, the contents may be received in real time. Inthis case, the streaming server may store the bitstream for apredetermined period of time to provide the streaming service smoothly.

For example, the user equipment may include a mobile phone, a smartphone, a laptop computer, a digital broadcasting terminal, a personaldigital assistant (PDA), a portable multimedia player (PMP), anavigation, a slate PC, a tablet PC, an ultrabook, a wearable device(e.g., a watch-type terminal (smart watch), a glass-type terminal (smartglass), a head mounted display (HMD)), a digital TV, a desktop computer,a digital signage or the like. Each of servers in the contents streamingsystem may be operated as a distributed server, and in this case, datareceived by each server may be processed in distributed manner.

FIG. 5 schematically illustrates a multiple transform techniqueaccording to an embodiment of the present disclosure.

Referring to FIG. 5, a transformer may correspond to the transformer inthe encoding apparatus of foregoing FIG. 2, and an inverse transformermay correspond to the inverse transformer in the encoding apparatus offoregoing FIG. 2, or to the inverse transformer in the decodingapparatus of FIG. 3.

The transformer may derive (primary) transform coefficients byperforming a primary transform based on residual samples (residualsample array) in a residual block (S510). This primary transform may bereferred to as a core transform. Herein, the primary transform may bebased on multiple transform selection (MTS), and when a multipletransform is applied as the primary transform, it may be referred to asa multiple core transform.

The multiple core transform may represent a method of transformingadditionally using discrete cosine transform (DCT) type 2 and discretesine transform (DST) type 7, DCT type 8, and/or DST type 1. That is, themultiple core transform may represent a transform method of transforminga residual signal (or residual block) of a space domain into transformcoefficients (or primary transform coefficients) of a frequency domainbased on a plurality of transform kernels selected from among the DCTtype 2, the DST type 7, the DCT type 8 and the DST type 1. Herein, theprimary transform coefficients may be called temporary transformcoefficients from the viewpoint of the transformer.

In other words, when the conventional transform method is applied,transform coefficients might be generated by applying transform from aspace domain to a frequency domain for a residual signal (or residualblock) based on the DCT type 2. Unlike to this, when the multiple coretransform is applied, transform coefficients (or primary transformcoefficients) may be generated by applying transform from a space domainto a frequency domain for a residual signal (or residual block) based onthe DCT type 2, the DST type 7, the DCT type 8, and/or DST type 1.Herein, the DCT type 2, the DST type 7, the DCT type 8, and the DST type1 may be called a transform type, transform kernel or transform core.These DCT/DST transform types can be defined based on basis functions.

When the multiple core transform is performed, a vertical transformkernel and a horizontal transform kernel for a target block may beselected from among the transform kernels, a vertical transform may beperformed on the target block based on the vertical transform kernel,and a horizontal transform may be performed on the target block based onthe horizontal transform kernel. Here, the horizontal transform mayindicate a transform on horizontal components of the target block, andthe vertical transform may indicate a transform on vertical componentsof the target block. The vertical transform kernel/horizontal transformkernel may be adaptively determined based on a prediction mode and/or atransform index for the target block (CU or subblock) including aresidual block.

Further, according to an example, if the primary transform is performedby applying the MTS, a mapping relationship for transform kernels may beset by setting specific basis functions to predetermined values andcombining basis functions to be applied in the vertical transform or thehorizontal transform. For example, when the horizontal transform kernelis expressed as trTypeHor and the vertical direction transform kernel isexpressed as trTypeVer, a trTypeHor or trTypeVer value of 0 may be setto DCT2, a trTypeHor or trTypeVer value of 1 may be set to DST7, and atrTypeHor or trTypeVer value of 2 may be set to DCT8.

In this case, MTS index information may be encoded and signaled to thedecoding apparatus to indicate any one of a plurality of transformkernel sets. For example, an MTS index of 0 may indicate that bothtrTypeHor and trTypeVer values are 0, an MTS index of 1 may indicatethat both trTypeHor and trTypeVer values are 1, an MTS index of 2 mayindicate that the trTypeHor value is 2 and the trTypeVer value. Is 1, anMTS index of 3 may indicate that the trTypeHor value is 1 and thetrTypeVer value is 2, and an MTS index of 4 may indicate that bothtrTypeHor and trTypeVer values are 2.

In one example, transform kernel sets according to MTS index informationare illustrated in the following table.

TABLE 1 tu_mts_idx[ x0 ][ y0 ] 0 1 2 3 4 trTypeHor 0 1 2 1 2 trTypeVer 01 1 2 2

The transformer may perform a secondary transform based on the (primary)transform coefficients to derive modified (secondary) transformcoefficients (S520). The primary transform is a transform from a spatialdomain to a frequency domain, and the secondary transform refers totransforming into a more compact expression using a correlation existingbetween (primary) transform coefficients. The secondary transform mayinclude a non-separable transform. In this case, the secondary transformmay be referred to as a non-separable secondary transform (NSST) or amode-dependent non-separable secondary transform (MDNSST). The NSST mayrepresent a transform that secondarily transforms (primary) transformcoefficients derived through the primary transform based on anon-separable transform matrix to generate modified transformcoefficients (or secondary transform coefficients) for a residualsignal. Here, the transform may be applied at once without separating(or independently applying a horizontal/vertical transform) a verticaltransform and a horizontal transform to the (primary) transformcoefficients based on the non-separable transform matrix. In otherwords, the NSST is not separately applied to the (primary) transformcoefficients in a vertical direction and a horizontal direction, and mayrepresent, for example, a transform method of rearrangingtwo-dimensional signals (transform coefficients) into a one-dimensionalsignal through a specific predetermined direction (e.g., row-firstdirection or column-first direction) and then generating modifiedtransform coefficients (or secondary transform coefficients) based onthe non-separable transform matrix. For example, a row-first order is todispose in a line in order of a 1st row, a 2nd row, . . . , an Nth rowfor M×N blocks, and a column-first order is to dispose in a line inorder of a 1st column, a 2nd column, . . . , an Mth column for M×Nblocks. The NSST may be applied to a top-left region of a block(hereinafter, referred to as a transform coefficient block) configuredwith (primary) transform coefficients. For example, when both a width Wand height H of the transform coefficient block are 8 or more, an 8×8NSST may be applied to the top-left 8×8 region of the transformcoefficient block. Further, while both the width (W) and height (H) ofthe transform coefficient block are 4 or more, when the width (W) orheight (H) of the transform coefficient block is smaller than 8, 4×4NSST may be applied to the top-left min(8,W)×min(8,H) region of thetransform coefficient block. However, the embodiment is not limitedthereto, and for example, even if only the condition that the width W orthe height H of the transform coefficient block is 4 or greater issatisfied, the 4×4 NSST may be applied to the top-left endmin(8,W)×min(8,H) region of the transform coefficient block.

Specifically, for example, if a 4×4 input block is used, thenon-separable secondary transform may be performed as follows.

The 4×4 input block X may be represented as follows.

$\begin{matrix}{X = \begin{bmatrix}X_{00} & X_{01} & X_{02} & X_{03} \\X_{10} & X_{11} & X_{12} & X_{13} \\X_{20} & X_{21} & X_{22} & X_{23} \\X_{30} & X_{31} & X_{32} & X_{33}\end{bmatrix}} & \left\lbrack {{Equation}1} \right\rbrack\end{matrix}$

If the X is represented in the form of a vector, the vector {right arrowover (X)} may be represented as below.

=[X ₀₀ X ₀₁ X ₀₂ X ₀₃ X ₁₀ X ₁₁ X ₁₂ X ₁₃ X ₂₀ X ₂₁ X ₂₂ X ₂₃ X ₃₀ X ₃₁X ₃₂ X ₃₃]^(T)  [Equation 2]

In Equation 2, the vector {right arrow over (X)} is a one-dimensionalvector obtained by rearranging the two-dimensional block X of Equation 1according to the row-first order.

In this case, the secondary non-separable transform may be calculated asbelow.

=T·

  [Equation 3]

In this equation, {right arrow over (F)} represents a transformcoefficient vector, and T represents a 16×16 (non-separable) transformmatrix.

Through foregoing Equation 3, a 16×1 transform coefficient vector {rightarrow over (F)} may be derived, and the {right arrow over (F)} may bere-organized into a 4×4 block through a scan order (horizontal,vertical, diagonal and the like). However, the above-describedcalculation is an example, and hypercube-Givens transform (HyGT) or thelike may be used for the calculation of the non-separable secondarytransform in order to reduce the computational complexity of thenon-separable secondary transform.

Meanwhile, in the non-separable secondary transform, a transform kernel(or transform core, transform type) may be selected to be modedependent. In this case, the mode may include the intra prediction modeand/or the inter prediction mode.

As described above, the non-separable secondary transform may beperformed based on an 8×8 transform or a 4×4 transform determined basedon the width (W) and the height (H) of the transform coefficient block.The 8×8 transform refers to a transform that is applicable to an 8×8region included in the transform coefficient block when both W and H areequal to or greater than 8, and the 8×8 region may be a top-left 8×8region in the transform coefficient block. Similarly, the 4×4 transformrefers to a transform that is applicable to a 4×4 region included in thetransform coefficient block when both W and H are equal to or greaterthan 4, and the 4×4 region may be a top-left 4×4 region in the transformcoefficient block. For example, an 8×8 transform kernel matrix may be a64×64/16×64 matrix, and a 4×4 transform kernel matrix may be a16×16/8×16 matrix.

Here, to select a mode-dependent transform kernel, two non-separablesecondary transform kernels per transform set for a non-separablesecondary transform may be configured for both the 8×8 transform and the4×4 transform, and there may be four transform sets. That is, fourtransform sets may be configured for the 8×8 transform, and fourtransform sets may be configured for the 4×4 transform. In this case,each of the four transform sets for the 8×8 transform may include two8×8 transform kernels, and each of the four transform sets for the 4×4transform may include two 4×4 transform kernels.

However, as the size of the transform, that is, the size of a region towhich the transform is applied, may be, for example, a size other than8×8 or 4×4, the number of sets may be n, and the number of transformkernels in each set may be k.

The transform set may be referred to as an NSST set or an LFNST set. Aspecific set among the transform sets may be selected, for example,based on the intra prediction mode of the current block (CU orsubblock). A low-frequency non-separable transform (LFNST) may be anexample of a reduced non-separable transform, which will be describedlater, and represents a non-separable transform for a low frequencycomponent.

For reference, for example, the intra prediction mode may include twonon-directional (or non-angular) intra prediction modes and 65directional (or angular) intra prediction modes. The non-directionalintra prediction modes may include a planar intra prediction mode of No.0 and a DC intra prediction mode of No. 1, and the directional intraprediction modes may include 65 intra prediction modes of Nos. 2 to 66.However, this is an example, and this document may be applied even whenthe number of intra prediction modes is different. Meanwhile, in somecases, intra prediction mode No. 67 may be further used, and the intraprediction mode No. 67 may represent a linear model (LM) mode.

FIG. 6 exemplarily shows intra directional modes of 65 predictiondirections.

Referring to FIG. 6, on the basis of intra prediction mode 34 having aleft upward diagonal prediction direction, the intra prediction modesmay be divided into intra prediction modes having horizontaldirectionality and intra prediction modes having verticaldirectionality. In FIG. 6, H and V denote horizontal directionality andvertical directionality, respectively, and numerals −32 to 32 indicatedisplacements in 1/32 units on a sample grid position. These numeralsmay represent an offset for a mode index value. Intra prediction modes 2to 33 have the horizontal directionality, and intra prediction modes 34to 66 have the vertical directionality. Strictly speaking, intraprediction mode 34 may be considered as being neither horizontal norvertical, but may be classified as belonging to the horizontaldirectionality in determining a transform set of a secondary transform.This is because input data is transposed to be used for a verticaldirection mode symmetrical on the basis of intra prediction mode 34, andan input data alignment method for a horizontal mode is used for intraprediction mode 34. Transposing input data means that rows and columnsof two-dimensional M×N block data are switched into N×M data. Intraprediction mode 18 and intra prediction mode 50 may represent ahorizontal intra prediction mode and a vertical intra prediction mode,respectively, and intra prediction mode 2 may be referred to as a rightupward diagonal intra prediction mode because intra prediction mode 2has a left reference pixel and performs prediction in a right upwarddirection. Likewise, intra prediction mode 34 may be referred to as aright downward diagonal intra prediction mode, and intra prediction mode66 may be referred to as a left downward diagonal intra prediction mode.

According to an example, the four transform sets according to the intraprediction mode may be mapped, for example, as shown in the followingtable.

TABLE 2 lfnstPredModeIntra lfnstTrSetIdx lfnstPredModeIntra < 0 1 0 <=lfnstPredModeIntra <= 1 0  2 <= lfnstPredModeIntra <= 12 1 13 <=lfnstPredModeIntra <= 23 2 24 <= lfnstPredModeIntra <= 44 3 45 <=lfnstPredModeIntra <= 55 2 56 <= lfnstPredModeIntra <= 80 1 81 <=lfnstPredModeIntra <= 83 0

As shown in Table 2, any one of the four transform sets, that is,lfnstTrSetldx, may be mapped to any one of four indexes, that is, 0 to3, according to the intra prediction mode.

When it is determined that a specific set is used for the non-separabletransform, one of k transform kernels in the specific set may beselected through a non-separable secondary transform index. An encodingapparatus may derive a non-separable secondary transform indexindicating a specific transform kernel based on a rate-distortion (RD)check and may signal the non-separable secondary transform index to adecoding apparatus. The decoding apparatus may select one of the ktransform kernels in the specific set based on the non-separablesecondary transform index. For example, lfnst index value 0 may refer toa first non-separable secondary transform kernel, lfnst index value 1may refer to a second non-separable secondary transform kernel, andlfnst index value 2 may refer to a third non-separable secondarytransform kernel. Alternatively, lfnst index value 0 may indicate thatthe first non-separable secondary transform is not applied to the targetblock, and lfnst index values 1 to 3 may indicate the three transformkernels.

The transformer may perform the non-separable secondary transform basedon the selected transform kernels, and may obtain modified (secondary)transform coefficients. As described above, the modified transformcoefficients may be derived as transform coefficients quantized throughthe quantizer, and may be encoded and signaled to the decoding apparatusand transferred to the dequantizer/inverse transformer in the encodingapparatus.

Meanwhile, as described above, if the secondary transform is omitted,(primary) transform coefficients, which are an output of the primary(separable) transform, may be derived as transform coefficientsquantized through the quantizer as described above, and may be encodedand signaled to the decoding apparatus and transferred to thedequantizer/inverse transformer in the encoding apparatus.

The inverse transformer may perform a series of procedures in theinverse order to that in which they have been performed in theabove-described transformer. The inverse transformer may receive(dequantized) transformer coefficients, and derive (primary) transformcoefficients by performing a secondary (inverse) transform (S550), andmay obtain a residual block (residual samples) by performing a primary(inverse) transform on the (primary) transform coefficients (S560). Inthis connection, the primary transform coefficients may be calledmodified transform coefficients from the viewpoint of the inversetransformer. As described above, the encoding apparatus and the decodingapparatus may generate the reconstructed block based on the residualblock and the predicted block, and may generate the reconstructedpicture based on the reconstructed block.

The decoding apparatus may further include a secondary inverse transformapplication determinator (or an element to determine whether to apply asecondary inverse transform) and a secondary inverse transformdeterminator (or an element to determine a secondary inverse transform).The secondary inverse transform application determinator may determinewhether to apply a secondary inverse transform. For example, thesecondary inverse transform may be an NSST, an RST, or an LFNST and thesecondary inverse transform application determinator may determinewhether to apply the secondary inverse transform based on a secondarytransform flag obtained by parsing the bitstream. In another example,the secondary inverse transform application determinator may determinewhether to apply the secondary inverse transform based on a transformcoefficient of a residual block.

The secondary inverse transform determinator may determine a secondaryinverse transform. In this case, the secondary inverse transformdeterminator may determine the secondary inverse transform applied tothe current block based on an LFNST (NSST or RST) transform setspecified according to an intra prediction mode. In an embodiment, asecondary transform determination method may be determined depending ona primary transform determination method. Various combinations ofprimary transforms and secondary transforms may be determined accordingto the intra prediction mode. Further, in an example, the secondaryinverse transform determinator may determine a region to which asecondary inverse transform is applied based on the size of the currentblock.

Meanwhile, as described above, if the secondary (inverse) transform isomitted, (dequantized) transform coefficients may be received, theprimary (separable) inverse transform may be performed, and the residualblock (residual samples) may be obtained. As described above, theencoding apparatus and the decoding apparatus may generate thereconstructed block based on the residual block and the predicted block,and may generate the reconstructed picture based on the reconstructedblock.

Meanwhile, in the present disclosure, a reduced secondary transform(RST) in which the size of a transform matrix (kernel) is reduced may beapplied in the concept of NSST in order to reduce the amount ofcomputation and memory required for the non-separable secondarytransform.

Meanwhile, the transform kernel, the transform matrix, and thecoefficient constituting the transform kernel matrix, that is, thekernel coefficient or the matrix coefficient, described in the presentdisclosure may be expressed in 8 bits. This may be a condition forimplementation in the decoding apparatus and the encoding apparatus, andmay reduce the amount of memory required to store the transform kernelwith a performance degradation that can be reasonably accommodatedcompared to the existing 9 bits or 10 bits. In addition, the expressingof the kernel matrix in 8 bits may allow a small multiplier to be used,and may be more suitable for single instruction multiple data (SIMD)instructions used for optimal software implementation.

In the present specification, the term “RST” may mean a transform whichis performed on residual samples for a target block based on a transformmatrix whose size is reduced according to a reduced factor. In the caseof performing the reduced transform, the amount of computation requiredfor transform may be reduced due to a reduction in the size of thetransform matrix. That is, the RST may be used to address thecomputational complexity issue occurring at the non-separable transformor the transform of a block of a great size.

RST may be referred to as various terms, such as reduced transform,reduced secondary transform, reduction transform, simplified transform,simple transform, and the like, and the name which RST may be referredto as is not limited to the listed examples. Alternatively, since theRST is mainly performed in a low frequency region including a non-zerocoefficient in a transform block, it may be referred to as aLow-Frequency Non-Separable Transform (LFNST). The transform index maybe referred to as an LFNST index.

Meanwhile, when the secondary inverse transform is performed based onRST, the inverse transformer 235 of the encoding apparatus 200 and theinverse transformer 322 of the decoding apparatus 300 may include aninverse reduced secondary transformer which derives modified transformcoefficients based on the inverse RST of the transform coefficients, andan inverse primary transformer which derives residual samples for thetarget block based on the inverse primary transform of the modifiedtransform coefficients. The inverse primary transform refers to theinverse transform of the primary transform applied to the residual. Inthe present disclosure, deriving a transform coefficient based on atransform may refer to deriving a transform coefficient by applying thetransform.

FIG. 7 is a diagram illustrating an RST according to an embodiment ofthe present disclosure.

In the present disclosure, a “target block” may refer to a current blockto be coded, a residual block, or a transform block.

In the RST according to an example, an N-dimensional vector may bemapped to an R-dimensional vector located in another space, so that thereduced transform matrix may be determined, where R is less than N. Nmay mean the square of the length of a side of a block to which thetransform is applied, or the total number of transform coefficientscorresponding to a block to which the transform is applied, and thereduced factor may mean an R/N value. The reduced factor may be referredto as a reduced factor, reduction factor, simplified factor, simplefactor or other various terms. Meanwhile, R may be referred to as areduced coefficient, but according to circumstances, the reduced factormay mean R. Further, according to circumstances, the reduced factor maymean the N/R value.

In an example, the reduced factor or the reduced coefficient may besignaled through a bitstream, but the example is not limited to this.For example, a predefined value for the reduced factor or the reducedcoefficient may be stored in each of the encoding apparatus 200 and thedecoding apparatus 300, and in this case, the reduced factor or thereduced coefficient may not be signaled separately.

The size of the reduced transform matrix according to an example may beR×N less than N×N, the size of a conventional transform matrix, and maybe defined as in Equation 4 below.

$\begin{matrix}{T_{R \times N} = \begin{bmatrix}t_{11} & t_{12} & t_{13} & \ldots & t_{1N} \\t_{21} & t_{22} & t_{23} & & t_{2N} \\ & \vdots & & \ddots & \vdots \\t_{R1} & t_{R2} & t_{R3} & \ldots & t_{RN}\end{bmatrix}} & \left\lbrack {{Equation}4} \right\rbrack\end{matrix}$

The matrix T in the Reduced Transform block shown in (a) of FIG. 7 maymean the matrix T_(R×N) of Equation 4. As shown in (a) of FIG. 7, whenthe reduced transform matrix T_(R×N) is multiplied to residual samplesfor the target block, transform coefficients for the target block may bederived.

In an example, if the size of the block to which the transform isapplied is 8×8 and R=16 (i.e., R/N=16/64=1/4), then the RST according to(a) of FIG. 7 may be expressed as a matrix operation as shown inEquation 5 below. In this case, memory and multiplication calculationcan be reduced to approximately ¼ by the reduced factor.

In the present disclosure, a matrix operation may be understood as anoperation of multiplying a column vector by a matrix, disposed on theleft of the column vector, to obtain a column vector.

$\begin{matrix}{\begin{bmatrix}t_{1,1} & t_{1,2} & t_{1,3} & \ldots & t_{1,64} \\t_{2,1} & t_{2,2} & t_{2,3} & & t_{2,64} \\ & \vdots & & \ddots & \vdots \\t_{16,1} & t_{16,2} & t_{16,3} & \ldots & t_{16,64}\end{bmatrix} \times \begin{bmatrix}r_{1} \\r_{2} \\ \vdots \\r_{64}\end{bmatrix}} & \left\lbrack {{Equation}5} \right\rbrack\end{matrix}$

In Equation 5, r₁ to r₆₄ may represent residual samples for the targetblock and may be specifically transform coefficients generated byapplying a primary transform. As a result of the calculation of Equation5 transform coefficients ci for the target block may be derived, and aprocess of deriving c_(i) may be as in Equation 6.

$\begin{matrix}{{{for}i{from}{to}R:}{{ci} = 0}{{for}j{from}1{to}N}{{ci}+={{tij}*{rj}}}} & \left\lbrack {{Equation}6} \right\rbrack\end{matrix}$

As a result of the calculation of Equation 6, transform coefficients c₁to c_(R) for the target block may be derived. That is, when R=16,transform coefficients c₁ to c₁₆ for the target block may be derived.If, instead of RST, a regular transform is applied and a transformmatrix of 64×64 (N×N) size is multiplied to residual samples of 64×1(N×1) size, then only 16 (R) transform coefficients are derived for thetarget block because RST was applied, although 64 (N) transformcoefficients are derived for the target block. Since the total number oftransform coefficients for the target block is reduced from N to R, theamount of data transmitted by the encoding apparatus 200 to the decodingapparatus 300 decreases, so efficiency of transmission between theencoding apparatus 200 and the decoding apparatus 300 can be improved.

When considered from the viewpoint of the size of the transform matrix,the size of the regular transform matrix is 64×64 (N×N), but the size ofthe reduced transform matrix is reduced to 16×64 (R×N), so memory usagein a case of performing the RST can be reduced by an R/N ratio whencompared with a case of performing the regular transform. In addition,when compared to the number of multiplication calculations N×N in a caseof using the regular transform matrix, the use of the reduced transformmatrix can reduce the number of multiplication calculations by the R/Nratio (R×N).

In an example, the transformer 232 of the encoding apparatus 200 mayderive transform coefficients for the target block by performing theprimary transform and the RST-based secondary transform on residualsamples for the target block. These transform coefficients may betransferred to the inverse transformer of the decoding apparatus 300,and the inverse transformer 322 of the decoding apparatus 300 may derivethe modified transform coefficients based on the inverse reducedsecondary transform (RST) for the transform coefficients, and may deriveresidual samples for the target block based on the inverse primarytransform for the modified transform coefficients.

The size of the inverse RST matrix T_(N×R) according to an example isN×R less than the size N×N of the regular inverse transform matrix, andis in a transpose relationship with the reduced transform matrix T_(R×N)shown in Equation 4.

The matrix Tt in the Reduced Inv. Transform block shown in (b) of FIG. 7may mean the inverse RST matrix T_(R×N) ^(T) (the superscript T meanstranspose). When the inverse RST matrix T_(R×N) ^(T) is multiplied tothe transform coefficients for the target block as shown in (b) of FIG.7, the modified transform coefficients for the target block or theresidual samples for the current block may be derived. The inverse RSTmatrix T_(R×N) ^(T) may be expressed as (T_(R×N)) T_(N×R).

More specifically, when the inverse RST is applied as the secondaryinverse transform, the modified transform coefficients for the targetblock may be derived when the inverse RST matrix T_(R×N) ^(T) ismultiplied to the transform coefficients for the target block.Meanwhile, the inverse RST may be applied as the inverse primarytransform, and in this case, the residual samples for the target blockmay be derived when the inverse RST matrix T_(R×N) ^(T) is multiplied tothe transform coefficients for the target block.

In an example, if the size of the block to which the inverse transformis applied is 8×8 and R=16 (i.e., R/N=16/64=1/4), then the RST accordingto (b) of FIG. 7 may be expressed as a matrix operation as shown inEquation 7 below.

$\begin{matrix}{\begin{bmatrix}t_{1,1} & & t_{2,1} & & t_{16,1} \\t_{1,2} & & t_{2,2} & \ldots & t_{16,2} \\t_{1,3} & & t_{2,3} & & t_{16,3} \\ \vdots & & \vdots & & \vdots \\ & \vdots & & \ddots & \vdots \\t_{1,64} & & t_{2,64} & \ldots & t_{16,64}\end{bmatrix} \times \begin{bmatrix}c_{1} \\c_{2} \\ \vdots \\c_{16}\end{bmatrix}} & \left\lbrack {{Equation}7} \right\rbrack\end{matrix}$

In Equation 7, c₁ to c₁₆ may represent the transform coefficients forthe target block. As a result of the calculation of Equation 7, r_(i)representing the modified transform coefficients for the target block orthe residual samples for the target block may be derived, and theprocess of deriving r_(i) may be as in Equation 8.

$\begin{matrix}{{{F{or}}i{from}1{to}N}{r_{i} = 0}{{for}j{from}1{to}R}{r_{i}+={t_{ji}*c_{j}}}} & \left\lbrack {{Equation}8} \right\rbrack\end{matrix}$

As a result of the calculation of Equation 8, r₁ to r_(N) representingthe modified transform coefficients for the target block or the residualsamples for the target block may be derived. When considered from theviewpoint of the size of the inverse transform matrix, the size of theregular inverse transform matrix is 64×64 (N×N), but the size of thereduced inverse transform matrix is reduced to 64×16 (R×N), so memoryusage in a case of performing the inverse RST can be reduced by an R/Nratio when compared with a case of performing the regular inversetransform. In addition, when compared to the number of multiplicationcalculations N×N in a case of using the regular inverse transformmatrix, the use of the reduced inverse transform matrix can reduce thenumber of multiplication calculations by the R/N ratio (N×R).

A transform set configuration shown in Table 2 may also be applied to an8×8 RST. That is, the 8×8 RST may be applied according to a transformset in Table 2. Since one transform set includes two or three transforms(kernels) according to an intra prediction mode, it may be configured toselect one of up to four transforms including that in a case where nosecondary transform is applied. In a transform where no secondarytransform is applied, it may be considered to apply an identity matrix.Assuming that indexes 0, 1, 2, and 3 are respectively assigned to thefour transforms (e.g., index 0 may be allocated to a case where anidentity matrix is applied, that is, a case where no secondary transformis applied), a transform index or an lfnst index as a syntax element maybe signaled for each transform coefficient block, thereby designating atransform to be applied. That is, for a top-left 8×8 block, through thetransform index, it is possible to designate an 8×8 RST in an RSTconfiguration, or to designate an 8×8 lfnst when the LFNST is applied.The 8×8 lfnst and the 8×8 RST refer to transforms applicable to an 8×8region included in the transform coefficient block when both W and H ofthe target block to be transformed are equal to or greater than 8, andthe 8×8 region may be a top-left 8×8 region in the transform coefficientblock. Similarly, a 4×4 lfnst and a 4×4 RST refer to transformsapplicable to a 4×4 region included in the transform coefficient blockwhen both W and H of the target block to are equal to or greater than 4,and the 4×4 region may be a top-left 4×4 region in the transformcoefficient block.

According to an embodiment of the present disclosure, for a transform inan encoding process, only 48 pieces of data may be selected and amaximum 16×48 transform kernel matrix may be applied thereto, ratherthan applying a 16×64 transform kernel matrix to 64 pieces of dataforming an 8×8 region. Here, “maximum” means that m has a maximum valueof 16 in an m×48 transform kernel matrix for generating m coefficients.That is, when an RST is performed by applying an m×48 transform kernelmatrix (m≤16) to an 8×8 region, 48 pieces of data are input and mcoefficients are generated. When m is 16, 48 pieces of data are inputand 16 coefficients are generated. That is, assuming that 48 pieces ofdata form a 48×1 vector, a 16×48 matrix and a 48×1 vector aresequentially multiplied, thereby generating a 16×1 vector. Here, the 48pieces of data forming the 8×8 region may be properly arranged, therebyforming the 48×1 vector. For example, a 48×1 vector may be constructedbased on 48 pieces of data constituting a region excluding the bottomright 4×4 region among the 8×8 regions. Here, when a matrix operation isperformed by applying a maximum 16×48 transform kernel matrix, 16modified transform coefficients are generated, and the 16 modifiedtransform coefficients may be arranged in a top-left 4×4 regionaccording to a scanning order, and a top-right 4×4 region and abottom-left 4×4 region may be filled with zeros.

For an inverse transform in a decoding process, the transposed matrix ofthe foregoing transform kernel matrix may be used. That is, when aninverse RST or LFNST is performed in the inverse transform processperformed by the decoding apparatus, input coefficient data to which theinverse RST is applied is configured in a one-dimensional vectoraccording to a predetermined arrangement order, and a modifiedcoefficient vector obtained by multiplying the one-dimensional vectorand a corresponding inverse RST matrix on the left of theone-dimensional vector may be arranged in a two-dimensional blockaccording to a predetermined arrangement order.

In summary, in the transform process, when an RST or LFNST is applied toan 8×8 region, a matrix operation of 48 transform coefficients intop-left, top-right, and bottom-left regions of the 8×8 region excludingthe bottom-right region among transform coefficients in the 8×8 regionand a 16×48 transform kernel matrix. For the matrix operation, the 48transform coefficients are input in a one-dimensional array. When thematrix operation is performed, 16 modified transform coefficients arederived, and the modified transform coefficients may be arranged in thetop-left region of the 8×8 region.

On the contrary, in the inverse transform process, when an inverse RSTor LFNST is applied to an 8×8 region, 16 transform coefficientscorresponding to a top-left region of the 8×8 region among transformcoefficients in the 8×8 region may be input in a one-dimensional arrayaccording to a scanning order and may be subjected to a matrix operationwith a 48×16 transform kernel matrix. That is, the matrix operation maybe expressed as (48×16 matrix)*(16×1 transform coefficient vector)=(48×1modified transform coefficient vector). Here, an n×1 vector may beinterpreted to have the same meaning as an n×1 matrix and may thus beexpressed as an n×1 column vector. Further, * denotes matrixmultiplication. When the matrix operation is performed, 48 modifiedtransform coefficients may be derived, and the 48 modified transformcoefficients may be arranged in top-left, top-right, and bottom-leftregions of the 8×8 region excluding a bottom-right region.

When a secondary inverse transform is based on an RST, the inversetransformer 235 of the encoding apparatus 200 and the inversetransformer 322 of the decoding apparatus 300 may include an inversereduced secondary transformer to derive modified transform coefficientsbased on an inverse RST on transform coefficients and an inverse primarytransformer to derive residual samples for the target block based on aninverse primary transform on the modified transform coefficients. Theinverse primary transform refers to the inverse transform of a primarytransform applied to a residual. In the present disclosure, deriving atransform coefficient based on a transform may refer to deriving thetransform coefficient by applying the transform.

The above-described non-separated transform, the LFNST, will bedescribed in detail as follows. The LFNST may include a forwardtransform by the encoding apparatus and an inverse transform by thedecoding apparatus.

The encoding apparatus receives a result (or a part of a result) derivedafter applying a primary (core) transform as an input, and applies aforward secondary transform (secondary transform).

y=G ^(T) x  [Equation 9]

In Equation 9, x and y are inputs and outputs of the secondarytransform, respectively, and G is a matrix representing the secondarytransform, and transform basis vectors are composed of column vectors.In the case of an inverse LFNST, when the dimension of thetransformation matrix G is expressed as [number of rows×number ofcolumns], in the case of an forward LFNST, the transposition of matrix Gbecomes the dimension of G^(T).

For the inverse LFNST, the dimensions of matrix G are [48×16], [48×8],[16×16], [16×8], and the [48×8] matrix and the [16×8] matrix are partialmatrices that sampled 8 transform basis vectors from the left of the[48×16] matrix and the [16×16] matrix, respectively.

On the other hand, for the forward LFNST, the dimensions of matrix G^(T)are [16×48], [8×48], [16×16], [8×16], and the [8×48] matrix and the[8×16] matrix are partial matrices obtained by sampling 8 transformbasis vectors from the top of the [16×48] matrix and the [16×16] matrix,respectively.

Therefore, in the case of the forward LFNST, a [48×1] vector or [16×1]vector is possible as an input x, and a [16×1] vector or a [8×1] vectoris possible as an output y. In video coding and decoding, the output ofthe forward primary transform is two-dimensional (2D) data, so toconstruct the [48×1] vector or the [16×1] vector as the input x, aone-dimensional vector must be constructed by properly arranging the 2Ddata that is the output of the forward transformation.

FIG. 8 is a diagram illustrating a sequence of arranging output data ofa forward primary transformation into a one-dimensional vector accordingto an example. The left diagrams of (a) and (b) of FIG. 8 show thesequence for constructing a [48×1] vector, and the right diagrams of (a)and (b) of FIG. 8 shows the sequence for constructing a [16×1] vector.In the case of the LFNST, a one-dimensional vector x can be obtained bysequentially arranging 2D data in the same order as in (a) and (b) ofFIG. 8.

The arrangement direction of the output data of the forward primarytransform may be determined according to an intra prediction mode of thecurrent block. For example, when the intra prediction mode of thecurrent block is in the horizontal direction with respect to thediagonal direction, the output data of the forward primary transform maybe arranged in the order of (a) of FIG. 8, and when the intra predictionmode of the current block is in the vertical direction with respect tothe diagonal direction, the output data of the forward primary transformmay be arranged in the order of (b) of FIG. 8.

According to an example, an arrangement order different from thearrangement orders of (a) and (b) FIG. 8 may be applied, and in order toderive the same result (y vector) as when the arrangement orders of (a)and (b) FIG. 8 is applied, the column vectors of the matrix G may berearranged according to the arrangement order. That is, it is possibleto rearrange the column vectors of G so that each element constitutingthe x vector is always multiplied by the same transform basis vector.

Since the output y derived through Equation 9 is a one-dimensionalvector, when two-dimensional data is required as input data in theprocess of using the result of the forward secondary transformation asan input, for example, in the process of performing quantization orresidual coding, the output y vector of Equation 9 must be properlyarranged as 2D data again.

FIG. 9 is a diagram illustrating a sequence of arranging output data ofa forward secondary transform into a two-dimensional block according toan example.

In the case of the LFNST, output values may be arranged in a 2D blockaccording to a predetermined scan order. (a) of FIG. 9 shows that whenthe output y is a [16×1] vector, the output values are arranged at 16positions of the 2D block according to a diagonal scan order. (b) ofFIG. 9 shows that when the output y is a [8×1] vector, the output valuesare arranged at 8 positions of the 2D block according to the diagonalscan order, and the remaining 8 positions are filled with zeros. X in(b) of FIG. 9 indicates that it is filled with zero.

According to another example, since the order in which the output vectory is processed in performing quantization or residual coding may bepreset, the output vector y may not be arranged in the 2D block as shownin FIG. 9. However, in the case of the residual coding, data coding maybe performed in 2D block (eg, 4×4) units such as CG (Coefficient Group),and in this case, the data are arranged according to a specific order asin the diagonal scan order of FIG. 9.

Meanwhile, the decoding apparatus may configure the one-dimensionalinput vector y by arranging two-dimensional data output through adequantization process or the like according to a preset scan order forthe inverse transformation. The input vector y may be output as theoutput vector x by the following equation.

X=Gy  [Equation 10]

In the case of the inverse LFNST, an output vector x can be derived bymultiplying an input vector y, which is a [16×1] vector or a [8×1]vector, by a G matrix. For the inverse LFNST, the output vector x can beeither a [48×1] vector or a [16×1] vector.

The output vector x is arranged in a two-dimensional block according tothe order shown in FIG. 8 and is arranged as two-dimensional data, andthis two-dimensional data becomes input data (or a part of input data)of the inverse primary transformation.

Accordingly, the inverse secondary transformation is the opposite of theforward secondary transformation process as a whole, and in the case ofthe inverse transformation, unlike in the forward direction, the inversesecondary transformation is first applied, and then the inverse primarytransformation is applied.

In the inverse LFNST, one of 8 [48×16] matrices and 8 [16×16] matricesmay be selected as the transformation matrix G. Whether to apply the[48×16] matrix or the [16×16] matrix depends on the size and shape ofthe block.

In addition, 8 matrices may be derived from four transform sets as shownin Table 2 above, and each transform set may consist of two matrices.Which transform set to use among the 4 transform sets is determinedaccording to the intra prediction mode, and more specifically, thetransform set is determined based on the value of the intra predictionmode extended by considering the Wide Angle Intra Prediction (WAIP).Which matrix to select from among the two matrices constituting theselected transform set is derived through index signaling. Morespecifically, 0, 1, and 2 are possible as the transmitted index value, 0may indicate that the LFNST is not applied, and 1 and 2 may indicate anyone of two transform matrices constituting a transform set selectedbased on the intra prediction mode value.

FIG. 10 is a diagram illustrating wide-angle intra prediction modesaccording to an embodiment of the present document.

The general intra prediction mode value may have values from 0 to 66 and81 to 83, and the intra prediction mode value extended due to WAIP mayhave a value from −14 to 83 as shown. Values from 81 to 83 indicate theCCLM (Cross Component Linear Model) mode, and values from −14 to −1 andvalues from 67 to 80 indicate the intra prediction mode extended due tothe WAIP application.

When the width of the prediction current block is greater than theheight, the upper reference pixels are generally closer to positionsinside the block to be predicted. Therefore, it may be more accurate topredict in the bottom-left direction than in the top-right direction.Conversely, when the height of the block is greater than the width, theleft reference pixels are generally close to positions inside the blockto be predicted. Therefore, it may be more accurate to predict in thetop-right direction than in the bottom-left direction. Therefore, it maybe advantageous to apply remapping, ie, mode index modification, to theindex of the wide-angle intra prediction mode.

When the wide-angle intra prediction is applied, information on theexisting intra prediction may be signaled, and after the information isparsed, the information may be remapped to the index of the wide-angleintra prediction mode. Therefore, the total number of the intraprediction modes for a specific block (e.g., a non-square block of aspecific size) may not change, and that is, the total number of theintra prediction modes is 67, and intra prediction mode coding for thespecific block may not be changed.

Table 3 below shows a process of deriving a modified intra mode byremapping the intra prediction mode to the wide-angle intra predictionmode.

TABLE 3 Inputs to this process are: - a variable predModeIntraspecifying the intra prediction mode, - a variable nTbW specifying thetransform block width, - a variable nTbH specifying the transform blockheight, - a variable cIdx specifying the colour component of the currentblock. Output of this process is the modified intra prediction modepredModeIntra. The variables nW and nH are derived as follows: - IfIntraSubPartitionsSplitType is equal to ISP_NO_SPLIT or cIdx is notequal to 0, the following applies:   nW = nTbW (8-97)   nH = nTbH(8-98) - Otherwise ( IntraSubPartitionsSplitType is not equal toISP_NO_SPLIT and cIdx is equal to 0 ), the following applies:   nW =nCbW (8-99)   nH = nCbH (8-100) The variable whRatio is set equal toAbs( Log2( nW / nH ) ). For non-square blocks (nW is not equal to nH),the intra prediction mode predModeIntra is modified as follows:  - Ifall of the following conditions are true, predModeIntra is set equal to( predModeIntra + 65 ).  - nW is greater than nH  - predModeIntra isgreater than or equal to 2  - predModeIntra is less than ( whRatio > 1 )? ( 8 + 2 * whRatio) : 8 - Otherwise, if all of the following conditionsare true, predModeIntra is set equal to  ( predModeIntra − 67 ).  - nHis greater than nW  - predModeIntra is less than or equal to 66  -predModeIntra is greater than ( whRatio > 1 ) ? ( 60 − 2 * whRatio ) :60

In Table 3, the extended intra prediction mode value is finally storedin the predModeIntra variable, and ISP_NO_SPLIT indicates that the CUblock is not divided into sub-partitions by the Intra Sub Partitions(ISP) technique currently adopted in the VVC standard, and the cIdxvariable Values of 0, 1, and 2 indicate the case of luma, Cb, and Crcomponents, respectively. Log 2 function shown in Table 3 returns a logvalue with a base of 2, and the Abs function returns an absolute value.

Variable predModeIntra indicating the intra prediction mode and theheight and width of the transform block, etc. are used as input valuesof the wide angle intra prediction mode mapping process, and the outputvalue is the modified intra prediction mode predModeIntra. The heightand width of the transform block or the coding block may be the heightand width of the current block for remapping of the intra predictionmode. At this time, the variable whRatio reflecting the ratio of thewidth to the width may be set to Abs(Log 2(nW/nH)).

For a non-square block, the intra prediction mode may be divided intotwo cases and modified.

First, if all conditions (1)˜(3) are satisfied, (1) the width of thecurrent block is greater than the height, (2) the intra prediction modebefore modifying is equal to or greater than 2, (3) the intra predictionmode is less than the value derived from (8+2*whRatio) when the variablewhRatio is greater than 1, and is less than 8 when the variable whRatiois less than or equal to 1 [predModeIntra is less than (whRatio>1)?(8+2*whRatio): 8], the intra prediction mode is set to a value 65greater than the intra prediction mode [predModeIntra is set equal to(predModeIntra+65)].

If different from the above, that is, follow conditions (1)˜(3) aresatisfied, (1) the height of the current block is greater than thewidth, (2) the intra prediction mode before modifying is less than orequal to 66, (3) the intra prediction mode is greater than the valuederived from (60−2*whRatio) when the variable whRatio is greater than 1,and is greater than 60 when the variable whRatio is less than or equalto 1 [predModeIntra is greater than (whRatio>1)? (60−2*whRatio): 60],the intra prediction mode is set to a value 67 smaller than the intraprediction mode [predModeIntra is set equal to (predModeIntra−67)].

Table 2 above shows how a transform set is selected based on the intraprediction mode value extended by the WAIP in the LFNST. As shown inFIG. 10, modes 14 to 33 and modes 35 to 80 are symmetric with respect tothe prediction direction around mode 34. For example, mode 14 and mode54 are symmetric with respect to the direction corresponding to mode 34.Therefore, the same transform set is applied to modes located inmutually symmetrical directions, and this symmetry is also reflected inTable 2.

Meanwhile, it is assumed that forward LFNST input data for mode 54 issymmetrical with the forward LFNST input data for mode 14. For example,for mode 14 and mode 54, the two-dimensional data is rearranged intoone-dimensional data according to the arrangement order shown in (a) ofFIG. 8 and (b) of FIG. 8, respectively. In addition, it can be seen thatthe patterns in the order shown in (a) of FIG. 8 and (b) of FIG. 8 aresymmetrical with respect to the direction (diagonal direction) indicatedby Mode 34.

Meanwhile, as described above, which transform matrix of the [48×16]matrix and the [16×16] matrix is applied to the LFNST is determined bythe size and shape of the transform target block.

FIG. 11 is a diagram illustrating a block shape to which the LFNST isapplied. (a) of FIG. 11 shows 4×4 blocks, (b) shows 4×8 and 8×4 blocks,(c) shows 4×N or N×4 blocks in which N is 16 or more, (d) shows 8×8blocks, (e) shows M×N blocks where M≥8, N≥8, and N>8 or M>8.

In FIG. 11, blocks with thick borders indicate regions to which theLFNST is applied. For the blocks of (a) and (b) of FIG. 11, the LFNST isapplied to the top-left 4×4 region, and for the block of (c) of FIG. 11,the LFNST is applied individually the two top-left 4×4 regions arecontinuously arranged. In (a), (b), and (c) of FIG. 11, since the LFNSTis applied in units of 4×4 regions, this LFNST will be hereinafterreferred to as “4×4 LFNST”. Based on the matrix dimension for G, a[16×16] or [16×8] matrix may be applied.

More specifically, the [16×8] matrix is applied to the 4×4 block (4×4 TUor 4×4 CU) of (a) of FIG. 11 and the [16×16] matrix is applied to theblocks in (b) and (c) of FIG. 11. This is to adjust the computationalcomplexity for the worst case to 8 multiplications per sample.

With respect to (d) and (e) of FIG. 11, the LFNST is applied to thetop-left 8×8 region, and this LFNST is hereinafter referred to as “8×8LFNST”. As a corresponding transformation matrix, a [48×16] matrix or[48×8] matrix may be applied. In the case of the forward LFNST, sincethe [48×1] vector (x vector in Equation 9) is input as input data, allsample values of the top-left 8×8 region are not used as input values ofthe forward LFNST. That is, as can be seen in the left order of (a) ofFIG. 8 or the left order of (b) of FIG. 8, the [48×1] vector may beconstructed based on samples belonging to the remaining 3 4×4 blockswhile leaving the bottom-right 4×4 block as it is.

The [48×8] matrix may be applied to an 8×8 block (8×8 TU or 8×8 CU) in(d) of FIG. 10, and the [48×16] matrix may be applied to the 8×8 blockin (e) of FIG. 11. This is also to adjust the computational complexityfor the worst case to 8 multiplications per sample.

Depending on the block shape, when the corresponding forward LFNST (4×4LFNST or 8×8 LFNST) is applied, 8 or 16 output data (y vector inEquation 9, [8×1] or [16×1] vector) is generated. In the forward LFNST,the number of output data is equal to or less than the number of inputdata due to the characteristics of the matrix GT.

FIG. 12 is a diagram illustrating an arrangement of output data of aforward LFNST according to an example, and shows a block in which outputdata of the forward LFNST is arranged according to a block shape.

The shaded area at the top-left of the block shown in FIG. 12corresponds to the area where the output data of the forward LFNST islocated, the positions marked with 0 indicate samples filled with avalue of 0, and the remaining area represents regions that are notchanged by the forward LFNST. In the area not changed by the LFNST, theoutput data of the forward primary transform remains unchanged.

As described above, since the dimension of the transform matrix appliedvaries according to the shape of the block, the number of output dataalso varies. As FIG. 12, the output data of the forward LFNST may notcompletely fill the top-left 4×4 block. In the case of (a) and (d) ofFIG. 12, a 16×8 matrix and a 48×8 matrix are applied to the blockindicated by a thick line or a partial region inside the block,respectively, and a 8×1 vector as the output of the forward LFNST isgenerated. That is, according to the scan order shown in (b) of FIG. 9,only 8 output data may be filled as shown in (a) and (d) of FIG. 12, and0 may be filled in the remaining 8 positions. In the case of the LFNSTapplied block of (d) of FIG. 11, as shown in (d) of FIG. 12, two 4×4blocks in the top-right and bottom-left adjacent to the top-left 4×4block are also filled with a value of 0.

As described above, basically, by signaling the LFNST index, whether toapply the LFNST and the transform matrix to be applied are specified. Asshown FIG. 12, when the LFNST is applied, since the number of outputdata of the forward LFNST may be equal to or less than the number ofinput data, a region filled with a zero value occurs as follows.

1) As shown in (a) of FIG. 12, samples from the 8th and later positionsin the scan order in the top-left 4×4 block, that is, samples from the9th to the 16th.

2) As shown in (d) and (e) of FIG. 12, when the 48×16 matrix or the 48×8matrix is applied, two 4×4 blocks adjacent to the top-left 4×4 block orthe second and third 4×4 blocks in the scan order.

Therefore, if non-zero data exists by checking the areas 1) and 2), itis certain that the LFNST is not applied, so that the signaling of thecorresponding LFNST index can be omitted.

According to an example, for example, in the case of LFNST adopted inthe VVC standard, since signaling of the LFNST index is performed afterthe residual coding, the encoding apparatus may know whether there isthe non-zero data (significant coefficients) for all positions withinthe TU or CU block through the residual coding. Accordingly, theencoding apparatus may determine whether to perform signaling on theLFNST index based on the existence of the non-zero data, and thedecoding apparatus may determine whether the LFNST index is parsed. Whenthe non-zero data does not exist in the area designated in 1) and 2)above, signaling of the LFNST index is performed.

Since a truncated unary code is applied as a binarization method for theLFNST index, the LFNST index consists of up to two bins, and 0, 10, and11 are assigned as binary codes for possible LFNST index values of 0, 1,and 2, respectively. In the case of the LFNST currently adopted for VVC,a context-based CABAC coding is applied to the first bin (regularcoding), and a bypass coding is applied to the second bin. The totalnumber of contexts for the first bin is 2, when (DCT-2, DCT-2) isapplied as a primary transform pair for the horizontal and verticaldirections, and a luma component and a chroma component are coded in adual tree type, one context is allocated and another context applies forthe remaining cases. The coding of the LFNST index is shown in a tableas follows.

TABLE 4 binIdx Syntax element 0 1 2 3 4 >=5 lfnst_idx[ ][ ] (tu_mts_idx[ x0 ][ y0 ] = = 0 bypass ns ns ns ns && treeType !=SINGLE_TREE ) ? 1 : 0

Meanwhile, for the adopted LFNST, the following simplification methodsmay be applied.

(i) According to an example, the number of output data for the forwardLFNST may be limited to a maximum of 16.

In the case of (c) of FIG. 11, the 4×4 LFNST may be applied to two 4×4regions adjacent to the top-left, respectively, and in this case, amaximum of 32 LFNST output data may be generated. when the number ofoutput data for forward LFNST is limited to a maximum of 16, in the caseof 4×N/N×4 (N≥16) blocks (TU or CU), the 4×4 LFNST is only applied toone 4×4 region in the top-left, the LFNST may be applied only once toall blocks of FIG. 11. Through this, the implementation of image codingmay be simplified.

FIG. 13 shows that the number of output data for the forward LFNST islimited to a maximum of 16 according to an example. As FIG. 13, when theLFNST is applied to the most top-left 4×4 region in a 4×N or N×4 blockin which N is 16 or more, the output data of the forward LFNST becomes16 pieces.

(ii) According to an example, zero-out may be additionally applied to aregion to which the LFNST is not applied. In this document, the zero-outmay mean filling values of all positions belonging to a specific regionwith a value of 0. That is, the zero-out can be applied to a region thatis not changed due to the LFNST and maintains the result of the forwardprimary transformation. As described above, since the LFNST is dividedinto the 4×4 LFNST and the 8×8 LFNST, the zero-out can be divided intotwo types ((ii)-(A) and (ii)-(B)) as follows.

(ii)-(A) When the 4×4 LFNST is applied, a region to which the 4×4 LFNSTis not applied may be zeroed out. FIG. 14 is a diagram illustrating thezero-out in a block to which the 4×4 LFNST is applied according to anexample.

As shown in FIG. 14, with respect to a block to which the 4×4 LFNST isapplied, that is, for all of the blocks in (a), (b) and (c) of FIG. 12,the whole region to which the LFNST is not applied may be filled withzeros.

On the other hand, (d) of FIG. 14 shows that when the maximum value ofthe number of the output data of the forward LFNST is limited to 16 asshown in FIG. 13, the zero-out is performed on the remaining blocks towhich the 4×4 LFNST is not applied.

(ii)-(B) When the 8×8 LFNST is applied, a region to which the 8×8 LFNSTis not applied may be zeroed out. FIG. 15 is a diagram illustrating thezero-out in a block to which the 8×8 LFNST is applied according to anexample.

As shown in FIG. 15, with respect to a block to which the 8×8 LFNST isapplied, that is, for all of the blocks in (d) and (e) of FIG. 12, thewhole region to which the LFNST is not applied may be filled with zeros.

(iii) Due to the zero-out presented in (ii) above, the area filled withzeros may be not same when the LFNST is applied. Accordingly, it ispossible to check whether the non-zero data exists according to thezero-out proposed in (ii) over a wider area than the case of the LFNSTof FIG. 12.

For example, when (ii)-(B) is applied, after checking whether thenon-zero data exists where the area filled with zero values in (d) and(e) of FIG. 12 in addition to the area filled with 0 additionally inFIG. 15, signaling for the LFNST index can be performed only when thenon-zero data does not exist.

Of course, even if the zero-out proposed in (ii) is applied, it ispossible to check whether the non-zero data exists in the same way asthe existing LFNST index signaling. That is, after checking whether thenon-zero data exists in the block filled with zeros in FIG. 12, theLFNST index signaling may be applied. In this case, the encodingapparatus only performs the zero out and the decoding apparatus does notassume the zero out, that is, checking only whether the non-zero dataexists only in the area explicitly marked as 0 in FIG. 12, may performthe LFNST index parsing.

Alternatively, according to another example, the zero-out may beperformed as shown in FIG. 16. FIG. 16 is a diagram illustrating thezero-out in a block to which the 8×8 LFNST is applied according toanother example.

As shown in FIGS. 14 and 15, the zero-out may be applied to all regionsother than the region to which the LFNST is applied, or the zero-out maybe applied only to a partial region as shown in FIG. 16. The zero-out isapplied only to regions other than the top-left 8×8 region of FIG. 16,the zero-out may not be applied to the bottom-right 4×4 block within thetop-left 8×8 region.

Various embodiments in which combinations of the simplification methods((i), (ii)-(A), (ii)-(B), (iii)) for the LFNST are applied may bederived. Of course, the combinations of the above simplification methodsare not limited to the following an embodiment, and any combination maybe applied to the LFNST.

EMBODIMENT

-   -   Limit the number of output data for forward LFNST to a maximum        of 16→(i)    -   When the 4×4 LFNST is applied, all areas to which the 4×4 LFNST        is not applied are zero-out→(ii)-(A)    -   When the 8×8 LFNST is applied, all areas to which the 8×8 LFNST        is not applied are zero-out→(ii)-(B)    -   After checking whether the non-zero data exists also the        existing area filled with zero value and the area filled with        zeros due to additional zero outs ((ii)-(A), (ii)-(B)), the        LFNST index is signaled only when the non-zero data does not        exist→(iii)

In the case of the Embodiment, when the LFNST is applied, an area inwhich the non-zero output data can exist is limited to the inside of thetop-left 4×4 area. In more detail, in the case of (a) of FIG. 14 and (a)of FIG. 15, the 8th position in the scan order is the last positionwhere non-zero data can exist. In the case of (b) and (c) of FIG. 14 and(b) of FIG. 15, the 16th position in the scan order (i.e., the positionof the bottom-right edge of the top-left 4×4 block) is the last positionwhere data other than 0 may exist.

Therefore, when the LFNST is applied, after checking whether thenon-zero data exists in a position where the residual coding process isnot allowed (at a position beyond the last position), it can bedetermined whether the LFNST index is signaled.

In the case of the zero-out method proposed in (ii), since the number ofdata finally generated when both the primary transform and the LFNST areapplied, the amount of computation required to perform the entiretransformation process can be reduced. That is, when the LFNST isapplied, since zero-out is applied to the forward primary transformoutput data existing in a region to which the LFNST is not applied,there is no need to generate data for the region that become zero-outduring performing the forward primary transform. Accordingly, it ispossible to reduce the amount of computation required to generate thecorresponding data. The additional effects of the zero-out methodproposed in (ii) are summarized as follows.

First, as described above, the amount of computation required to performthe entire transform process is reduced.

In particular, when (ii)-(B) is applied, the amount of calculation forthe worst case is reduced, so that the transform process can belightened. In other words, in general, a large amount of computation isrequired to perform a large-size primary transformation. By applying(ii)-(B), the number of data derived as a result of performing theforward LFNST can be reduced to 16 or less. In addition, as the size ofthe entire block (TU or CU) increases, the effect of reducing the amountof transform operation is further increased.

Second, the amount of computation required for the entire transformprocess can be reduced, thereby reducing the power consumption requiredto perform the transform.

Third, the latency involved in the transform process is reduced.

The secondary transformation such as the LFNST adds a computationalamount to the existing primary transformation, thus increasing theoverall delay time involved in performing the transformation. Inparticular, in the case of intra prediction, since reconstructed data ofneighboring blocks is used in the prediction process, during encoding,an increase in latency due to a secondary transformation leads to anincrease in latency until reconstruction. This can lead to an increasein overall latency of intra prediction encoding.

However, if the zero-out suggested in (ii) is applied, the delay time ofperforming the primary transform can be greatly reduced when LFNST isapplied, the delay time for the entire transform is maintained orreduced, so that the encoding apparatus can be implemented more simply.

In the conventional intra prediction, a block to be currently encoded isregarded as one encoding unit and encoding was performed withoutsplitting. However, intra sub-partitions (ISP) coding means performingintra prediction encoding by dividing a block to be currently encoded ina horizontal direction or a vertical direction. In this case, areconstructed block may be generated by performing encoding/decoding inunits of divided blocks, and the reconstructed block may be used as areference block of the next divided block. According to an embodiment,in ISP coding, one coding block may be divided into two or foursub-blocks and coded, and in ISP, in one sub-block, intra prediction isperformed with reference to a reconstructed pixel value of a sub-blocklocated at the adjacent left side or adjacent upper side. Hereinafter,“coding” may be used as a concept including both coding performed by anencoding apparatus and decoding performed by a decoding apparatus.

Table 5 represents the number of sub-blocks divided according to blocksizes when ISP is applied, and sub-partitions divided according to ISPmay be referred to as transform blocks (TUs).

TABLE 5 Block size (CU) Number of divisions 4 × 4 Not available 4 × 8, 8× 4 2 All other cases 4

The ISP is to divide a block predicted as luma intra into two or foursub-partitions in a vertical direction or a horizontal directionaccording to the size of the block. For example, the minimum block sizeto which the ISP can be applied is 4×8 or 8×4. When the block size isgreater than 4×8 or 8×4, the block is divided into 4 sub-partitions.

FIGS. 17 and 18 illustrate an example of a sub-block into which onecoding block is divided, and more specifically, FIG. 17 illustrates anexample of division of a case in which a coding block (width (W)×height(H)) is 4×8 blocks or 8×4 blocks, and FIG. 18 illustrates an example ofdivision of a case in which a coding block is not 4×8 blocks, 8×4blocks, or 4×4 blocks.

When ISP is applied, sub-blocks are sequentially coded, for example,horizontally or vertically, from left to right or from top to bottomaccording to a division type, and after a reconstruction process isperformed via inverse transform and intra prediction for one sub-block,coding for the next sub-block may be performed. For the leftmost oruppermost subblock, a reconstructed pixel of the already coded codingblock is referred to, as in a conventional intra prediction method.Further, when each side of a subsequent internal sub-block is notadjacent to a previous sub-block, in order to derive reference pixelsadjacent to the corresponding side, reconstructed pixels of an alreadycoded adjacent coding block are referred to, as in a conventional intraprediction method.

In the ISP coding mode, all sub-blocks may be coded with the same intraprediction mode, and a flag indicating whether to use ISP coding and aflag indicating whether to divide (horizontally or vertically) in whichdirection may be signaled. As illustrated in FIGS. 17 and 18, the numberof sub-blocks may be adjusted to 2 or 4 according to a block shape, andwhen the size (width×height) of one sub-block is less than 16, it may berestricted so that division into the corresponding sub-block is notallowed or the ISP coding itself is not applied.

In the case of the ISP prediction mode, one coding unit is divided intotwo or four partition blocks, that is, sub-blocks and predicted, and thesame intra prediction mode is applied to the divided two or fourpartition blocks.

As described above, in the division direction, the horizontal direction(when an M×N coding unit having horizontal and vertical lengths of M andN, respectively, is divided in the horizontal direction, if the M×Ncoding unit is divided into two, the M×N coding unit is divided intoM×(N/2) blocks, and if the M×N coding unit is divided into four blocks,the M×N coding unit is divided into M×(N/4) blocks) and the verticaldirection (when an M×N coding unit is divided in a vertical direction,if the M×N coding unit is divided into two, and the M×N coding unit isdivided into (M/2)×N blocks, and if the M×N coding unit is divided intofour, the M×N coding unit is divided into (M/4)×N blocks) are bothpossible. When the M×N coding unit is divided in the horizontaldirection, partition blocks are coded in a top-down order, and when theM×N coding unit is divided in the vertical direction, partition blocksare coded in order from left to right. The currently coded partitionblock may be predicted with reference to the reconstructed pixel valuesof the upper (left) partition block in the case of horizontal (vertical)direction division.

A transform may be applied to a residual signal generated by the ISPprediction method in units of partition blocks. Multiple transformselection (MTS) technology based on a DST-7/DCT-8 combination as well asthe existing DCT-2 may be applied to a primary transform (coretransform) based on a forward direction, and a forward low frequencynon-separable transform (LFNST) may be applied to transform coefficientsgenerated according to the primary transform to generate a finalmodified transform coefficient.

That is, an LFNST may be applied to partition blocks divided by applyingan ISP prediction mode, and the same intra prediction mode is applied tothe divided partition blocks, as described above. Accordingly, when theLFNST set derived based on the intra prediction mode is selected, thederived LFNST set may be applied to all partition blocks. That is,because the same intra prediction mode is applied to all partitionblocks, the same LFNST set may be applied to all partition blocks.

According to an embodiment, an LFNST may be applied only to transformblocks having both a horizontal length and a vertical length of 4 ormore. Therefore, when the horizontal or vertical length of the dividedpartition block according to the ISP prediction method is less than 4,the LFNST is not applied and an LFNST index is not signaled. Further,when the LFNST is applied to each partition block, the correspondingpartition block may be regarded as one transform block. When the ISPprediction method is not applied, the LFNST may be applied to the codingblock.

A method of applying an LFNST to each partition block will be describedin detail.

According to an embodiment, after a forward LFNST is applied toindividual partition blocks, only maximum 16 (8 or 16) coefficients areleft in the top-left 4×4 region in transform coefficient scanning order,and then zero out in which the remaining positions and regions are allfilled with a value of 0 may be applied.

Alternatively, according to an embodiment, when a length of one side ofthe partition block is 4, the LFNST is applied only to the top-left 4×4region, and when a length of all sides of the partition block, that is,the width and height is 8 or more, the LFNST may be applied to theremaining 48 coefficients, except for a bottom-right 4×4 region inside atop-left 8×8 region.

Alternatively, according to an embodiment, in order to adjustcomputational complexity of the worst case to 8 multiplications persample, when each partition block is 4×4 or 8×8, only 8 transformcoefficients may be output after applying the forward LFNST. That is,when the partition block is 4×4, an 8×16 matrix may be applied as atransform matrix, and when the partition block is 8×8, an 8×48 matrixmay be applied as a transform matrix.

In the current VVC standard, LFNST index signaling is performed in unitsof coding units. Therefore, in the ISP prediction mode and when an LFNSTis applied to all partition blocks, the same LFNST index value may beapplied to the corresponding partition blocks. That is, when the LFNSTindex value is transmitted once at a coding unit level, thecorresponding LFNST index may be applied to all partition blocks in thecoding unit. As described above, the LFNST index value may have valuesof 0, 1, and 2, where 0 represents a case in which an LFNST is notapplied, and 1 and 2 indicate two transform matrices existing in oneLFNST set when an LFNST is applied.

As described above, the LFNST set is determined by the intra predictionmode, and in the case of the ISP prediction mode, because all partitionblocks in the coding unit are predicted in the same intra predictionmode, the partition blocks may refer to the same LFNST set.

As another example, LFNST index signaling is still performed in units ofa coding unit, but in the case of the ISP prediction mode, whether touniformly apply an LFNST to all partition blocks is not determined, andfor each partition block, whether to apply the LFNST index valuesignaled at a coding unit level or whether not to apply the LFNST may bedetermined through a separate condition. Here, a separate condition maybe signaled in the form of a flag for each partition block through abitstream, and when a flag value is 1, an LFNST index value signaled atthe coding unit level is applied, and when a flag value is 0, the LFNSTmay not be applied.

In a coding unit to which the ISP mode is applied, an example ofapplying an LFNST when a length of one side of the partition block isless than 4 is described as follows.

First, when the size of the partition block is N×2 (2×N), an LFNST maybe applied to the top-left M×2 (2×M) region (where M≤N). For example,when M=8, the top-left region becomes 8×2 (2×8) and thus a region inwhich 16 residual signals exist may be an input of a forward LFNST, andan R×16 (R≤16) forward transform matrix may be applied.

Here, the forward LFNST matrix may be a separate additional matrix otherthan the matrix included in the current VVC standard. Further, forcomplexity control of the worst case, an 8×16 matrix in which only upper8 row vectors of a 16×16 matrix are sampled may be used fortransformation. The complexity control method will be described indetail later.

Second, when the size of the partition block is N×1 (1×N), an LFNST maybe applied to a top-left M×1 (1×M) region (where M≤N). For example, whenM=16, the top-left region becomes 16×1 (1×16) and thus a region in which16 residual signals exist may be an input of the forward LFNST, and anR×16 (R≤16) forward transform matrix may be applied.

Here, the corresponding forward LFNST matrix may be a separateadditional matrix other than the matrix included in the current VVCstandard. Further, for complexity control of the worst case, an 8×16matrix in which only the upper 8 row vectors of the 16×16 matrix aresampled may be used for transformation. The complexity control methodwill be described in detail later.

The first embodiment and the second embodiment may be appliedsimultaneously, or either one of the two embodiments may be applied. Inparticular, in the case of the second embodiment, because a primarytransform is considered in an LFNST, it was observed through experimentsthat compression performance improvement that could be obtained in theexisting LFNST is relatively small compared to an LFNST index signalingcost. However, in the case of the first embodiment, compressionperformance improvement similar to that which could be obtained from theconventional LFNST was observed. That is, in the case of ISP, it may bechecked through experiments that the application of LFNST for 2×N andN×2 contributes to an actual compression performance.

In an LFNST in the current VVC, symmetry between intra prediction modesis applied. The same LFNST set is applied to two directional modesdisposed around a mode 34 (prediction in a 45 degree diagonal directionto the bottom-right), for example, the same LFNST set is applied to amode 18 (horizontal direction prediction mode) and a mode 50 (verticaldirection prediction mode). However, in modes 35 to 66, when a forwardLFNST is applied, input data is transposed and then an LFNST is applied.

VVC supports a wide angle intra prediction (WAIP) mode, and an LFNST setis derived based on the modified intra prediction mode in considerationof the WAIP mode. For modes extended by WAIP, the LFNST set isdetermined by using symmetry, as in the general intra predictiondirection mode. For example, because a mode −1 is symmetric with a mode67, the same LFNST set is applied, and because a mode −14 is symmetricwith a mode 80, the same LFNST set is applied. Modes 67 to 80 apply anLFNST transform after transposing input data before applying a forwardLFNST.

In the case of the LFNST applied to the top-left M×2 (M×1) block,because the block to which the LFNST is applied is non-square, thesymmetry to the LFNST cannot be applied. Therefore, instead of applyingthe symmetry based on the intra prediction mode, as in the LFNST ofTable 2, the symmetry between the M×2 (M×1) block and the 2×M (1×M)block may be applied.

FIG. 19 is a diagram illustrating symmetry between M×2 (M×1) blocks and2×M (1×M) blocks according to an embodiment.

As illustrated in FIG. 19, because it may be regarded that a mode 2 inthe M×2 (M×1) block is symmetric with a mode 66 in the 2×M (1×M) block,the same LFNST set may be applied to the 2×M (1×M) block and the M×2(M×1) block.

In this case, in order to apply the LFNST set applied to the M×2 (M×1)block to the 2×M (1×M) block, the LFNST set is selected based on themode 2 instead of the mode 66. That is, before applying the forwardLFNST, after transposing input data of the 2×M (1×M) block, the LFNSTmay be applied.

FIG. 20 is a diagram illustrating an example of transposing a 2×M blockaccording to an embodiment.

(a) of FIG. 20 is a diagram illustrating that an LFNST may be applied byreading input data in column-first order for a 2×M block, and (b) ofFIG. 20 is a diagram illustrating that an LFNST may be applied byreading input data in row-first order for an M×2 (M×1) block. A methodof applying an LFNST to the top-left M×2 (M×1) or 2×M (M×1) block isdescribed as follows.

1. First, as illustrated in (a) and (b) of FIG. 20, input data isarranged to configure an input vector of the forward LFNST. For example,referring to FIG. 19, for an M×2 block predicted in a mode 2, the orderin (b) of FIG. 20 is followed, and for a 2×M block predicted in a mode66, the input data are arranged in order of (a) of FIG. 20 and then theLFNST set for the mode 2 may be applied.

2. For the M×2 (M×1) block, the LFNST set is determined based on amodified intra prediction mode in consideration of WAIP. As describedabove, a preset mapping relationship is established between the intraprediction mode and the LFNST set, which may be represented by a mappingtable as shown in table 2.

For a 2×M (1×M) block, a symmetric mode around a prediction mode (mode34 in the case of the VVC standard) in a 45 degree diagonal directiondownward from the modified intra prediction mode in consideration ofWAIP is obtained, and then the LFNST set is determined based on thecorresponding symmetric mode and the mapping table. A symmetrical mode(y) around the mode 34 may be derived through the following equation.The mapping table will be described in more detail below.

if 2≤x≤66, y=68−x,

otherwise (x≤−1 or x≥67), y=66−x  [Equation 11]

3. When a forward LFNST is applied, transform coefficients may bederived by multiplying input data prepared in a process 1 by an LFNSTkernel. An LFNST kernel may be selected from an LFNST set determined ina process 2 and a predetermined LFNST index.

For example, when M=8 and a 16×16 matrix is applied as the LFNST kernel,16 transform coefficients may be generated by multiplying the matrix by16 input data. The generated transform coefficients may be arranged inthe top-left 8×2 or 2×8 region in scanning order used in the VVCstandard.

FIG. 21 illustrates scanning order for 8×2 or 2×8 regions according toan embodiment.

All regions other than the top-left 8×2 or 2×8 region may be filled withzero values (zero-out) or the existing transform coefficients to which aprimary transform is applied may be maintained as they are. Thepredetermined LFNST index may be one of the LFNST index values (0, 1, 2)attempted when calculating an RD cost while changing the LFNST indexvalue in an encoding process.

In the case (e.g., 8 multiplications/samples) of a configuration thatadjusts computational complexity for the worst case to a certain levelor less, for example, after generating only 8 transform coefficients bymultiplying an 8×16 matrix that takes only the upper 8 rows of the 16×16matrix, 8 transform coefficients may be disposed in scanning order ofFIG. 21, and zero-out may be applied to the remaining coefficientregions. The complexity control for the worst case will be describedlater.

4. When applying an inverse LFNST, a preset number (e.g., 16) oftransform coefficients are set as input vectors, and the LFNST setobtained from a process 2 and the LFNST kernel (e.g., 16×16 matrix)derived from the parsed LFNST index are selected and then by multiplyingthe LFNST kernel and the corresponding input vector, the output vectormay be derived.

In the case of an M×2 (M×1) block, the output vectors may be disposed inrow-first order of (b) of FIG. 20, and in the case of the 2×M (1×M)block, the output vectors may be disposed in column-first order of (a)of FIG. 20.

The remaining regions, except for a region in which the correspondingoutput vector is disposed within the top-left M×2 (M×1) or 2×M (M×2)region and a region other than the top-left M×2 (M×1) or 2×M (M×2)region in the partition block may be all zero-out with zero values ormay be configured to maintain reconstructed transform coefficients asthey are through residual coding and inverse quantization processes.

When configuring the input vector, as in No. 3, input data may bearranged in scanning order of FIG. 21, and in order to controlcomputational complexity for the worst case to a certain level or less,an input vector may be configured by reducing the number of input data(e.g., 8 instead of 16).

For example, when M=8, if 8 input data are used, only the left 16×8matrix may be taken from the corresponding 16×16 matrix and multipliedto obtain 16 output data. The complexity control for the worst case willbe described later.

In the above embodiment, when applying LFNST, a case in which symmetryis applied between an M×2 (M×1) block and a 2×M (1×M) block ispresented, but according to another example, different LFNST sets may beapplied to each of two block shapes.

Hereinafter, various examples of an LFNST set configuration for the ISPmode and a mapping method using the intra prediction mode will bedescribed.

In the case of an ISP mode, the LFNST set configuration may be differentfrom the existing LFNST set. In other words, kernels different from theexisting LFNST kernels may be applied, and a mapping table differentfrom the mapping table between an intra prediction mode index applied tothe current VVC standard and the LFNST set may be applied. A mappingtable applied to the current VVC standard may be the same as that ofTable 2.

In Table 2, a preModeIntra value means an intra prediction mode valuechanged in consideration of WAIP, and an lfnstTrSetIdx value is an indexvalue indicating a specific LFNST set. Each LFNST set is configured withtwo LFNST kernels.

When the ISP prediction mode is applied, if both a horizontal length anda vertical length of each partition block are equal to or greater than4, the same kernels as the LFNST kernels applied in the current VVCstandard may be applied, and the mapping table may be applied as it is.A mapping table and LFNST kernels different from the current VVCstandard may be applied.

When the ISP prediction mode is applied, when a horizontal length or avertical length of each partition block is less than 4, a mapping tableand LFNST kernels different from those in the current VVC standard maybe applied. Hereinafter, Tables 6 to 8 represent mapping tables betweenan intra prediction mode value (intra prediction mode value changed inconsideration of WAIP) and an LFNST set, which may be applied to an M×2(M×1) block or a 2×M (1×M) block.

TABLE 6 predModeIntra lfnstTrSetIdx predModeIntra < 0 1 0 <=predModeIntra <= 1 0  2 <= predModeIntra <= 12 1 13 <= predModeIntra <=23 2 24 <= predModeIntra <= 34 3 35 <= predModeIntra <= 44 4 45 <=predModeIntra <= 55 5 56 <= predModeIntra <= 66 6 67 <= predModeIntra <=80 6 81 <= predModeIntra <= 83 0

TABLE 7 predModeIntra lfnstTrSetIdx predModeIntra < 0 1 0 <=predModeIntra <= 1 0  2 <= predModeIntra <= 23 1 24 <= predModeIntra <=44 2 45 <= predModeIntra <= 66 3 67 <= predModeIntra <= 80 3 81 =predModeIntra <= 83 0

TABLE 8 predModeIntra lfnstTrSetIdx predModeIntra < 0 1 0 <=predModeIntra <= 1 0  2 <= predModeIntra <= 80 1 81 = predModeIntra <=83 0

A first mapping table of Table 6 is configured with seven LFNST sets, amapping table of Table 7 is configured with four LFNST sets, and amapping table of Table 8 is configured with two LFNST sets. As anotherexample, when it is configured with one LFNST set, an lfnstTrSetIdxvalue may be fixed to 0 with respect to the preModeIntra value.

Hereinafter, a method of maintaining the computational complexity forthe worst case when LFNST is applied to the ISP mode will be described.

In the case of an ISP mode, when LFNST is applied, in order to maintainthe number of multiplications per sample (or per coefficient, perposition) to a certain value or less, the application of LFNST may belimited. According to the size of the partition block, the number ofmultiplications per sample (or per coefficient, per position) may bemaintained to 8 or less by applying LFNST as follows.

1. When both a horizontal length and a vertical length of the partitionblock are 4 or more, the same method as a calculation complexity controlmethod for the worst case for LFNST in the current VVC standard may beapplied.

That is, when the partition block is a 4×4 block, instead of a 16×16matrix, an 8×16 matrix obtained by sampling the top 8 rows from a 16×16matrix may be applied in a forward direction, and a 16×8 matrix obtainedby sampling the left 8 columns from a 16×16 matrix may be applied in areverse direction. Further, when the partition block is 8×8 blocks, inthe forward direction, instead of a 16×48 matrix, an 8×48 matrixobtained by sampling the top 8 rows from a 16×48 matrix is applied, andin the reverse direction, instead of a 48×16 matrix, a 48×8 matrixobtained by sampling the left 8 columns from a 48×16 matrix may beapplied.

In the case of a 4×N or N×4 (N>4) block, when a forward transform isperformed, 16 coefficients generated after applying a 16×16 matrix toonly the top-left 4×4 block may be disposed in the top-left 4×4 region,and other regions may be filled with a value of 0. Further, whenperforming an inverse transform, 16 coefficients located in the top-left4×4 block are disposed in scanning order to form an input vector, andthen 16 output data may be generated by multiplying the 16×16 matrix.The generated output data may be disposed in the top-left 4×4 region,and the remaining regions, except for the top-left 4×4 region, may befilled with a value of 0.

In the case of an 8×N or N×8 (N>8) block, when the forward transform isperformed, 16 coefficients generated after applying the 16×48 matrix toonly an ROI region (the remaining regions excluding bottom-right 4×4blocks from the top-left 8×8 blocks) inside the top-left 8×8 blocks maybe disposed in the top-left 4×4 region, and all other regions may befilled with a value of 0. Further, when performing an inverse transform,16 coefficients located in the top-left 4×4 block are disposed inscanning order to form an input vector, and then 48 output data may begenerated by multiplying the input vector by a 48×16 matrix. Thegenerated output data may be filled in the ROI region, and all otherregions may be filled with a value of 0.

2. When the size of the partition block is N×2 or 2×N and LFNST isapplied to the top-left M×2 or 2×M region (M≤N), a matrix sampledaccording to the N value may be applied.

In the case of M=8, for a partition block of N=8, that is, an 8×2 or 2×8block, an 8×16 matrix obtained by sampling the top 8 rows from a 16×16matrix may be applied instead of a 16×16 matrix in the case of a forwardtransform, and in the case of an inverse transform, instead of a 16×16matrix, a 16×8 matrix obtained by sampling left 8 columns from the 16×16matrix may be applied.

When N is greater than 8, 16 output data generated after applying the16×16 matrix to the top-left 8×2 or 2×8 block in the case of a forwardtransform are disposed in the top-left 8×2 or 2×8 block, and theremaining regions may be filled with a value of 0. In the case of aninverse transform, 16 coefficients located in the top-left 8×2 or 2×8block are disposed in the scanning order to form an input vector, andthen 16 output data may be generated by multiplying the 16×16 matrix.The generated output data may be disposed in the top-left 8×2 or 2×8block, and all remaining regions may be filled with a value of 0.

3. When the size of the partition block is N×1 or 1×N and LFNST isapplied to the top-left M×1 or 1×M region (M≤N), a matrix sampledaccording to the N value may be applied.

When M=16, for a partition block of N=16, that is, a 16×1 or 1×16 block,instead of a 16×16 matrix, an 8×16 matrix obtained by sampling the top 8rows from a 16×16 matrix may be applied in the case of a forwardtransform, and in the case of an inverse transform, instead of a 16×16matrix, a 16×8 matrix obtained by sampling the left 8 columns from the16×16 matrix may be applied.

When N is greater than 16, 16 output data generated after applying the16×16 matrix to the top-left 16×1 or 1×16 block in the case of a forwardtransform may be disposed in the top-left 16×1 or 1×16 block, and theremaining regions may be filled with a value of 0. In the case of aninverse transform, 16 coefficients located in the top-left 16×1 or 1×16block may be disposed in scanning order to form an input vector, andthen 16 output data may be generated by multiplying the 16×16 matrix.The generated output data may be disposed in the top-left 16×1 or 1×16block, and all remaining regions may be filled with a value of 0.

As another example, in order to maintain the number of multiplicationsper sample (or per coefficient, per position) to a certain value orless, the number of multiplications per sample (or per coefficient, perposition) based on the ISP coding unit size rather than the size of theISP partition block may be maintained to 8 or less. When there is onlyone block satisfying the condition to which LFNST is applied among theISP partition blocks, the complexity calculation for the worst case ofLFNST may be applied based on the corresponding coding unit size ratherthan the size of the partition block. For example, a luma coding blockfor one coding unit (CU) is divided (or partitioned) to four partitionblocks each having a size of 4×4. And, herein, among the four partitionblocks, if a non-zero transform coefficient does not exist for twopartition blocks, each of the remaining two partition blocks may beconfigured to have 16 transform coefficients, instead of 8 transformcoefficients, generated therein (based on the encoder).

Hereinafter, a method for signaling an LFNST index in case of an ISPmode will be described.

As described above, an LFNST index may have a value of 0, 1, 2, wherein0 indicates that LFNST is not applied, and wherein 1 and 2 respectivelyindicate each one of two LFNST kernel matrices that are included in aselected LFNST set. LFNST is applied based on an LFNST kernel matrixthat is selected by the LFNST index. In the current VVC standard, amethod according to which an LFNST is transmitted will be described asfollows.

1. An LFNST index may be transmitted once for each coding unit (CU),and, in case of a dual-tree, an LFNST index may be separately signaledfor each of a luma block and a chroma block.

2. When an LFNST index is not signaled, the LFNST index is inferred as0, which is a default value. Cases where the LFNST index value isinferred as 0 will be described below.

A. When the mode corresponds to a mode in which transform is not applied(e.g., transform skip, BDPCM, lossless coding, and so on)

B. When a primary transform is not DCT-2 (DST7 or DCT8), i.e., when ahorizontal transform or a vertical transform is not DCT-2

C. When a horizontal length or vertical length of a luma block of acoding unit exceeds a maximum luma transform size that is available fortransform, e.g., when a maximum luma transform size that is availablefor transform is equal to 64, and when the size of a luma block of acoding block is equal to 128×16, LFNST cannot be applied.

In case of a dual-tree, it is determined whether or not the maximum lumatransform size is exceeded for each of a coding unit for a luma elementand a coding unit for a chroma element. That is, it is checked whetheror not the maximum luma transform size that is available for transformis exceeded for a luma block, and it is checked whether or nothorizontal/vertical lengths and a maximum luma transform size that isavailable for transform of a corresponding luma block for a color formatare exceeded for a chroma block. For example, when the color format is4:2:0, each of the horizontal/vertical lengths of the corresponding lumablock becomes two times the lengths of the corresponding chroma block,and the transform size of the corresponding luma block becomes 2 timesthe size of the corresponding chroma block. As another example, when thecolor format is 4:4:4, the horizontal/vertical lengths and transformsize of the corresponding luma block are the same as the correspondingchroma block.

A 64-length transform or 32-length transform means a transform beingapplied to a horizontal or vertical length of 64 or 32, respectively.And, a “transform size” may mean the corresponding length of 64 or 32.

In case of a single-tree, after checking whether or not the horizontallength or vertical length for a luma block exceeds the maximum lumatransform block size that is available for transform, when the lengthexceeds the transform block size, LFNST index signaling may be skipped(or omitted).

D. An LFNST index may be transmitted only when both the horizontallength and the vertical length of a coding unit is equal to 4 or more.

In case of a dual-tree, an LFNST index may be signaled only when boththe horizontal length and the vertical length for the correspondingelement (i.e., luma element or chroma element) is equal to 4 or more.

In case of a single-tree, an LFNST index may be signaled when both thehorizontal length and the vertical length for the luma element is equalto 4 or more.

E. When a last non-zero coefficient position is not a DC position(top-left position in a block), if the block is a dual-tree type lumablock, and if the last non-zero coefficient position is not a DCposition, an LFNST index is transmitted. If the block is a dual-treetype chroma block, and if at least one of a last non-zero coefficientposition for Cb and a last non-zero coefficient position for Cr is not aDC position, a corresponding LFNST index is transmitted.

In case of a single-tree type, for any one of a luma element, a Cbelement, and a Cr element, if a corresponding last non-zero coefficientposition is not a DC position, an LFNST index is transmitted.

Herein, when a coded block flag (CBF) value, which indicates thepresence or absence of a transform coefficient for one transform block,is equal to 0, in order to determine whether or not to perform LFNSTindex signaling, the last non-zero coefficient position for thecorresponding transform block is not checked. That is, when thecorresponding CBF value is equal to 0, since transform is not applied tothe corresponding block, when checking the conditions for LFNST indexsignaling, the last non-zero coefficient position may not be considered.

For example, 1) in case of a dual-tree type and a luma element, if acorresponding CBF value is equal to 0, an LFNST index is not signaled,2) in case of a dual-tree type and a chroma element, if a CBF value forCb is equal to 0 and a CBF value for Cr is equal to 1, only the positionof the last non-zero coefficient position for Cr is checked so as totransmit the corresponding LFNST index, and 3) in case of a single-treetype, only the last non-zero coefficient position(s) for the lumaelement, Cb element, or Cr element each having a CBF value of 1 is/arechecked.

F. When it is verified that a transform coefficient exists in a positionother than a position where an LFNST transform coefficient may exist,LFNST index signaling may be skipped (or omitted). In case of a 4×4transform block and an 8×8 transform block, according to a transformcoefficient scanning order of a VVC standard, the LFNST transformcoefficient may exist in 8 positions starting from a DC position, andall of the remaining positions may be filled with 0s. Additionally, in acase where the transform block is not a 4×4 transform block and an 8×8transform block, according to a transform coefficient scanning order ofa VVC standard, the LFNST transform coefficient may exist in 16positions starting from a DC position, and all of the remainingpositions may be filled with 0s.

Therefore, after carrying out residual coding, when a non-zero transformcoefficient exists in a region that should only be filled with the 0value, LFNST index signaling may be skipped (or omitted).

Meanwhile, the ISP mode may be applied only to a luma block or may beapplied to both luma block and chroma block. As described above, whenISP prediction is applied, prediction is carried out after dividing (orpartitioning) a corresponding coding unit to 2 or 4 partition blocks,and the transform may also be applied to each of the correspondingpartition blocks. Therefore, even when determining the conditions forsignaling an LFNST index by coding units, it should be considered thatthe LFNST may be applied to each of the corresponding partition blocks.Additionally, when the ISP prediction mode is applied only to a specificelement (e.g., luma block), the LFNST index should be signaled based onthe fact that the coding unit is divided into partition blocks only forthe corresponding element. LFNST index signaling methods that areavailable for the ISP mode will be described below.

1. An LFNST index may be transmitted once for each coding unit (CU),and, in case of a dual-tree, an LFNST index may be separately signaledfor each of a luma block and a chroma block.

2. When an LFNST index is not signaled, the LFNST index is inferred as0, which is a default value. Cases where the LFNST index value isinferred as 0 will be described below.

A. When the mode corresponds to a mode in which transform is not applied(e.g., transform skip, BDPCM, lossless coding, and so on)

B. When a horizontal length or vertical length of a luma block of acoding unit exceeds a maximum luma transform size that is available fortransform, e.g., when a maximum luma transform size that is availablefor transform is equal to 64, and when the size of a luma block of acoding block is equal to 128×16, LFNST cannot be applied.

Whether or not to perform signaling of an LFNST index may be determinedbased on a size of a partition block instead of a coding unit. That is,when the horizontal length or vertical length of a partition block forthe corresponding luma block exceeds the maximum luma transform sizethat is available for transform, the LFNST index signaling may beskipped (or omitted), and the LFNST index value may be inferred as 0.

In case of a dual-tree, it is determined whether or not the maximumblock transform size is exceeded for each coding unit or partition blockfor a luma element and for each coding unit or partition block for achroma element. That is, by comparing each of the horizontal length andthe vertical length of the coding unit or partition block for the lumaelement with the maximum luma transform size, and when it is determinedthat at least one length is larger than the maximum luma transform size,the LFNST is not applied. And, in case of the coding unit orpartitioning block for the chroma element, the horizontal/verticallengths of a corresponding luma block for a color format are compared tothe maximum luma transform size that is available for maximum transform.For example, when the color format is 4:2:0, each of thehorizontal/vertical lengths of the corresponding luma block becomes twotimes the lengths of the corresponding chroma block, and the transformsize of the corresponding luma block becomes 2 times the size of thecorresponding chroma block. As another example, when the color format is4:4:4, the horizontal/vertical lengths and transform size of thecorresponding luma block are the same as the corresponding chroma block.

In case of a single-tree, after checking whether or not the horizontallength or vertical length for a luma block (coding unit or partitionblock) exceeds the maximum luma transform block size that is availablefor transform, when the length exceeds the transform block size, LFNSTindex signaling may be skipped (or omitted).

C. If LFNST that is included in the current VVC standard is applied, anLFNST index may be transmitted only when both the horizontal length andthe vertical length of a partition block is equal to 4 or more.

Apart from the LFNST that is included in the current VVC standard, ifLFNST for a 2×M (1×M) or M×2 (M×1) block is applied, an LFNST index maybe transmitted only for a case where the partition block size is equalto or larger than the 2×M (1×M) or M×2 (M×1) block. Herein, when a P×Qblock is equal to or larger than an R×S block, this means that P≥R andQ≥S.

In summary, an LFNST index may be transmitted only for a case where apartition block size is equal to or larger than a minimum size to whichLFNST can be applied. In case of the dual-tree, an LFNST index may besignaled only in a case where the size of a partition block for a lumaor chroma element is equal to or larger than a minimum size to whichLFNST can be applied. In case of the single-tree, an LFNST index may besignaled only in a case where the size of a partition block for a lumaelement is equal to or larger than a minimum size to which LFNST can beapplied.

In the present specification, when an M×N block is equal to or largerthan a K×L block, this means that M is equal to or larger than K andthat N is equal to or larger than L. When an M×N block is larger than aK×L block, this means that M is equal to or larger than K and that N isequal to or larger than L, while M is larger than K or N is larger thanL. When an M×N block is smaller than or equal to a K×L block, this meansthat M is less than or equal to K and that N is less than or equal to L.And, when an M×N block is smaller than a K×L block, this means that M isless than or equal to K and that N is less than or equal to L, while Mis less than K or N is less than L.

D. When a last non-zero coefficient position is not a DC position(top-left position in a block), if the block is a dual-tree type lumablock, and if a corresponding last non-zero coefficient position foreven one of all partition blocks is not a DC position, an LFNST indexmay be transmitted. If the block is a dual-tree type chroma block, andif even one of a last non-zero coefficient position of all partitionblocks for Cb (when the ISP mode is not applied to the chroma element,it is given that the number of partition blocks is equal to 1) and alast non-zero coefficient position of all partition blocks for Cr (whenthe ISP mode is not applied to the chroma element, it is given that thenumber of partition blocks is equal to 1) is not a DC position, acorresponding LFNST index may be transmitted.

In case of a single-tree type, for any one of a luma element, a Cbelement, and a Cr element, if a corresponding last non-zero coefficientposition for even one of all partition blocks is not a DC position, anLFNST index may be transmitted.

Herein, when a coded block flag (CBF) value, which indicates thepresence or absence of a transform coefficient for each partition block,is equal to 0, in order to determine whether or not to perform LFNSTindex signaling, the last non-zero coefficient position for thecorresponding partition block is not checked. That is, when thecorresponding CBF value is equal to 0, since transform is not applied tothe corresponding block, when checking the conditions for LFNST indexsignaling, the last non-zero coefficient position for the correspondingpartition block is considered.

For example, 1) in case of a dual-tree type and a luma element, if acorresponding CBF value for each partition block is equal to 0, thecorresponding partition block is excluded when determining whether ornot to perform LFNST index signaling, 2) in case of a dual-tree type anda chroma element, if a CBF value for Cb is equal to 0 and a CBF valuefor Cr is equal to 1 for each partition block, only the position of thelast non-zero coefficient position for Cr is checked, so as to determinewhether or not to perform the corresponding LFNST index signaling, and3) in case of a single-tree type, only the last non-zero coefficientposition(s) for the luma element, Cb element, or Cr element each havinga CBF value of 1 for all partition blocks are checked, so as todetermine whether or not to perform the LFNST index signaling.

In case of the ISP mode, image information may be configured so that thelast non-zero coefficient position is not checked, and the correspondingembodiments will be described below.

i. In case of the ISP mode, checking of the last non-zero coefficientposition for both luma block and chroma block is skipped, and LFNSTindex signaling may be authorized. That is, even if the last non-zerocoefficient position for all partition blocks is the DC position or hasa corresponding CBF value of 0, the corresponding LFNST index signalingmay be authorized.

ii. In case of the ISP mode, checking of the last non-zero coefficientposition for only the luma block is skipped, and, for the chroma block,checking of the last non-zero coefficient position according to theabove-described method may be performed. For example, in case of thedual-tree type and the luma block, checking of the last non-zerocoefficient position is not performed, and the LFNST index signaling maybe authorized. And, in case of the dual-tree type and the chroma block,the presence or absence of a DC position corresponding to the lastnon-zero coefficient position is checked according to theabove-described method, so as to determine whether or not to performsignaling of the corresponding LFNST index.

iii. In case of the ISP mode and the single-tree type, the method numberi and the method number ii may be applied. That is, in case of applyingthe method number i to the ISP mode and the single-tree type, checkingof the last non-zero coefficient position for both the luma block andthe chroma block may be skipped, and the LFNST index signaling may beauthorized. Alternatively, by applying the method number ii, checking ofthe last non-zero coefficient position for the partition blocks of theluma element may be skipped, and checking of the last non-zerocoefficient position for the partition blocks of the chroma element(when the ISP mode is not applied to the chroma element, it may be giventhat the number of partition blocks is equal to 1) may be performedaccording to the above-described method, so as to determine whether ornot to perform the corresponding LFNST index signaling.

E. When it is verified that a transform coefficient exists in a positionother than a position where an LFNST transform coefficient may existeven for one partition block among all of the partition blocks, theLFNST index signaling may be skipped (or omitted).

For example, in case of a 4×4 partition block and an 8×8 partitionblock, according to a transform coefficient scanning order of a VVCstandard, the LFNST transform coefficient may exist in 8 positionsstarting from a DC position, and all of the remaining positions may befilled with 0s. Additionally, in a case where the partition block isequal to or greater than 4×4, and in case the partition block is not a4×4 partition block and an 8×8 partition block, according to a transformcoefficient scanning order of a VVC standard, the LFNST transformcoefficient may exist in 16 positions starting from a DC position, andall of the remaining positions may be filled with 0s.

Therefore, after carrying out residual coding, when a non-zero transformcoefficient exists in a region that should only be filled with the 0value, LFNST index signaling may be skipped (or omitted).

If LFNST can be applied even for a case where the partition block sizeis equal to 2×M (1×M) or M×2 (M×1), a region where the LFNST transformcoefficient can be positioned may be designated as described below. Aregion outside of the region where the LFNST transform coefficient canbe positioned may be filled with 0s. And, when it is assumed that LFNSThas been applied, if a non-zero transform coefficient exists in theregion that should be filled with 0s, the LFNST index signaling may beskipped.

i. When LFNST can be applied to a 2×M or M×2 block, and when M=8, only 8LFNST transform coefficients may be generated for a 2×8 or 8×2 partitionblock. When the transform coefficients are arranged in a scanning orderthat is shown in FIG. 20, 8 transform coefficients are arranged in thescanning order starting from the DC position, and the remaining 8positions may be filled with 0s.

16 LFNST transform coefficients may be generated for a 2×N or N×2 (N>8)partition block. When the transform coefficients are arranged in thescanning order that is shown in FIG. 20, 16 transform coefficients arearranged in the scanning order starting from the DC position, and theremaining region may be filled with 0s. That is, in the 2×N or N×2 (N>8)partition block, a region excluding the top-left 2×8 or 8×2 block may befilled with 0s. Instead of 8 LFNST transform coefficients, 16coefficient blocks may also be generated for a 2×8 or 8×2 partitionblock, and, in this case, there is no region that needs to be filledwith 0s. As described above, when the LFNST is applied, when a non-zerotransform coefficient is detected to exist in a region that isdesignated to be filled with 0s even in one partition block, the LFNSTindex signaling may be skipped, and the LFNST index may be inferred as0.

ii. When LFNST can be applied to a 1×M or M×1 block, and when M=16, only8 LFNST transform coefficients may be generated for a 1×16 or 16×1partition block. When the transform coefficients are arranged in aleft-to-right or top-to-bottom scanning order, 8 transform coefficientsare arranged in the corresponding scanning order starting from the DCposition, and the remaining 8 positions may be filled with 0s.

16 LFNST transform coefficients may be generated for a 1×N or N×1 (N>16)partition block. When the transform coefficients are arranged in theleft-to-right or top-to-bottom scanning order, 16 transform coefficientsare arranged in the corresponding scanning order starting from the DCposition, and the remaining region may be filled with 0s. That is, inthe 1×N or N×1 (N>16) partition block, a region excluding the top-left1×16 or 16×1 block may be filled with 0s.

Instead of 8 LFNST transform coefficients, 16 coefficient blocks mayalso be generated for a 1×16 or 16×1 partition block, and, in this case,there is no region that needs to be filled with 0s. As described above,when the LFNST is applied, when a non-zero transform coefficient isdetected to exist in a region that is designated to be filled with 0seven in one partition block, the LFNST index signaling may be skipped,and the LFNST index may be inferred as 0.

Meanwhile, in case of the ISP mode, in the current VVC standard, byindependently (or separately) referring to the length condition for thehorizontal direction and the vertical direction, DST-7 is applied,instead of DCT-2, without performing signaling for an MTS index.Depending upon whether or not the horizontal or vertical length is equalto or greater than 4 and less than or equal to 16, a primary transformkernel is determined. Therefore, in case of the ISP mode, and when LFNSTmay be applied, the following transform combination may be configured asdescribed below.

1. For a case where the LFNST index is 0 (including a case where theLFNST index is inferred as 0), a condition for determining a primarytransform corresponding to the ISP mode that is included in the currentVVC standard may be followed. That is, by independently (or separately)checking whether or not the length condition (i.e., the condition of thelength being equal to or greater than 4 and less than or equal to 16)for the horizontal direction and the vertical direction is satisfied, ifthe length condition is satisfied, DST-7 is applied for the primarytransform, instead of DCT-2. And, if the length condition is notsatisfied, DCT-2 may be applied.

2. For a case where the LFNST index is greater than 0, the following twoconfigurations may be possible for the primary transform.

A. DCT-2 may be applied for both the horizontal direction and verticaldirection.

B. A condition for determining a primary transform corresponding to theISP mode that is included in the current VVC standard may be followed.That is, by independently (or separately) checking whether or not thelength condition (i.e., the condition of the length being equal to orgreater than 4 and less than or equal to 16) for the horizontaldirection and the vertical direction is satisfied, if the lengthcondition is satisfied, DST-7 is applied, instead of DCT-2. And, if thelength condition is not satisfied, DCT-2 may be applied.

In case of the ISP mode, image information may be configured so that anLFNST index can be transmitted for each partition block, instead ofbeing transmitted for each coding unit.

In this case, the above-described LFNST index signaling method assumesthat only one partition block exists within the unit through which theLFNST index is being transmitted, and whether or not to perform LFNSTindex signaling may be determined.

Meanwhile, an embodiment of an LFNST kernel that may be applied to a 2×Nor N×2 (N≥8) partition block, when in the ISP mode, will hereinafter beshown. The following LFNST kernel may be applied to a top-left 2×8 or8×2 region in the 2×N or N×2 (N≥8) partition block. And, thecorresponding LFNST kernel may be used in the LFNST that is described inthe above-described embodiments.

Each of Table 9 to Table 11 shows an example in which LFNST kernels areconfigured of a total of 4 LFNST sets, wherein each LFNST set isconfigured of two LFNST kernel candidates, i.e., an LFNST kernel matrix.This is a method of selecting one set among 4 LFNST sets, as shown inTable 2. The LFNST kernels that are proposed in Table 9 to Table 11 aredefined according to the grammar of C/C++ programming languages. And, ing_lfnst_2×8_8×2[4][2][16][16], which is an array that stores LFNSTkernel data, [4] indicates that the kernels are configured of a total of4 LFNST sets, [2] indicates that each LFNST set is configured of 2 LFNSTkernel candidates, and [16][16] indicates that each LFNST kernel isconfigured of a 16×16 matrix. Each 16×16 matrix shown in Table 9 toTable 11 represents a matrix that is used in forward LFNST transform.That is, one row becomes one transform basis vector (1×16 vector), whichis then multiplied by input data that is configured of primary transformcoefficients.

TABLE 9 const int8_t g_lfnst_2×8_8×2[4][2][16][16] = {  { //0   {    {99,−21,−70,20,14,−7,−3,2,−18,4,10,−3,−1,0,0,2 },    {2,−90,48,63,−36,−6,10,−1,−15,17,2,−12,4,4,−4,0 },    {55,14,59,−64,−43,33,3,−8,−46,10,20,−3,2,2,−1 },    {42,−6,19,−11,−25,9,2,3,80,−13,−75,23,25,−12,−5,3 },    {−9,−73,5,−56,72,30,−37,5,0,9,−9,12,−6,−8,4,0 },    {33,28,64,16,56,−75,−23,26,−7,4,−7,2,−5,6,5,−4 },    {−6,8,−12,−19,7,−8,11,1,0,79,−32,−76,40,20,−21,−1 },    {12,34,21,55,16,73,−70,−16,15,10,9,−22,0,2,−2,6 },    {−11,12,−5,17,8,6,−8,−11,−74,−7,−51,37,65,−39,−11,15 },    {−12,−8,−21,−16,−51,−18,−74,80,−8,−16,0,−11,13,8,−6,0 },    {−5,10,−9,6,−19,−3,−9,16,3,86,−7,43,−42,−52,42,4 },    {4,−3,9,−6,6,−8,8,11,35,10,69,4,46,−56,−39,56 },    {4,10,9,19,27,53,59,91,−6,2,−1,10,4,11,−3,−12 },    {−1,1,−3,1,−4,−3,−13,−12,12,37,30,69,36,51,−39,−58 },    {−3,3,−2,2,−3,1,−1,2,−13,13,−26,25,−55,39,−71,72 },    {−2,−2,−1,−1,0,0,1,−1,5,8,10,21,39,64,75,66 },   },   {    {100,39,−56,−36,−3,3,0,1,−15,−5,7,4,0,−1,1,2 },    {20,−85,−57,50,50,4,−11,1,−8,9,8,−1,−4,−4,−2,0 },    {−68,36,−77,−39,41,27,4,−1,21,−5,4,2,−6,−6,0,0 },    {20,−1,−6,4,−5,−5,4,7,94,43,−56,−43,−4,9,2,0 },    {−1,70,−5,76,37,−54,−34,1,3,−14,0,−1,1,3,3,1 },    {22,10,57,−22,81,52,−40,−28,3,15,0,−2,−10,−4,4,0 },    {5,−14,0,−6,9,7,−14,−3,13,−71,−69,43,60,11,−15,−3 },    {−7,−26,7,−67,30,−87,−35,33,−11,6,−8,−7,−7,5,9,1 },    {−11,9,−11,3,5,11,7,−14,−70,20,−50,−66,25,50,12,−12 },    {10,10,21,5,55,−9,99,49,−5,6,−5,13,11,−2,−7,−3 },    {−6,9,−8,−1,−5,−13,−9,−23,−17,84,−14,46,52,−37,−39,−12 },    {−4,9,0,12,−14,44,−44,103,−17,16,−14,3,5,−16,8,16 },    {−3,1,−3,4,−2,−7,15,−23,−28,−3,−64,10,−50,−68,36,47 },    {−2,0,−4,4,−2,2,2,−5,7,33,−2,66,−3,64,81,12 },    {0,−3,3,−5,5,−4,5,−9,11,−1,35,−32,72,−15,29,87 },    {2,−2,1,0,0,0,1,1,7,−9,10,−18,37,−55,75,−77 },   }  },  { //1   {    {−112,34,23,−2,5,0,2,0,44,−11,−9,0,−2,0,0,0 },    {23,96,−70,6,−12,1,−4,1,18,−33,11,2,1,1,0,1 },    {−21,42,25,−49,14,−11,6,−3,−100,−7,23,9,6,0,2,0 },    {46,29,62,−73,2,−6,−1,−2,49,−12,−37,19,−3,1,0,1 },    {12,31,63,62,−77,6,−16,−1,−20,−10,−15,−16,16,4,4,2 },    {−5,−42,1,−16,−25,28,3,1,3,−106,37,20,6,2,−3,2 },    {16,7,28,50,67,−73,1,−8,2,−48,5,−14,−14,17,3,2 },    {−8,−6,−22,17,21,22,−25,2,−33,−28,−100,44,−1,−2,7,2 },    {−10,−16,−27,−15,−51,−71,73,−11,−4,−4,−35,8,18,16,−11,−1 },    {10,15,16,24,29,57,80,−63,−1,−2,−4,0,−1,−16,−18,8 },    {−1,−7,−11,−26,12,11,−5,−3,−5,−23,−33,−106,46,−7,13,4 },    {−7,−10,−12,−15,−20,−20,−52,−107,1,3,3,−1,−19,7,11,20 },    {−2,1,−2,−10,−15,18,14,15,−12,−7,−15,−40,−101,54,−10,3 },    {0,−1,0,−1,−14,−17,12,3,−4,−10,−5,−11,−52,−101,47,−19 },    {−1,−1,0,−1,−3,−14,−17,12,−5,−4,−5,−9,−11,−48,−97,61 },    {0,0,1,2,0,2,16,18,3,5,5,8,2,8,62,108 },   },   {    {123,−10,−22,−8,−4,−3,−2,−1,−22,4,2,1,1,0,0,0 },    {−6,−118,28,30,3,5,4,1,−22,17,2,−4,0,0,−1,−1 },    {20,−24,−4,5,1,0,0,1,122,0,−22,−5,−4,−2,−3,0 },    {−21,−22,−117,6,39,6,1,5,−4,5,11,−2,−4,−2,0,0 },    {0,24,1,23,0,−4,0,−1,1,120,−10,−25,−3,−4,−3,−2 },    {−10,−26,−3,−118,−18,23,9,3,1,28,−5,0,0,−2,1,0 },    {8,4,32,−15,77,6,−14,−5,17,10,89,7,−21,−3,−1,−3 },    {4,2,19,−16,89,−10,−20,−1,−14,−3,−83,6,14,0,1,2 },    {6,10,4,20,6,123,4,−22,−1,−1,−12,5,3,−4,−1,1 },    {4,5,7,−3,21,2,74,8,−4,−20,−1,−98,−6,14,0,−1 },    {−3,−4,−3,−6,−14,9,−100,−2,−3,−13,1,−74,−12,8,5,1 },    {3,4,6,5,3,20,−8,124,1,1,4,4,18,0,−1,1 },    {−1,−1,−1,−2,4,−3,−6,−18,8,2,23,−14,123,−1,−13,−1 },    {0,0,−2,1,1,3,−2,−1,3,8,1,15,3,126,9,−4 },    {0,1,0,−2,−1,0,−4,1,−4,−3,−4,0,−12,9,−124,−28 },    {0,0,0,0,0,0,0,−1,0,2,2,0,−3,6,−28,125 },   }  },  { //2   {    {112,−10,−39,−1,−1,−2,−1,0,−45,10,15,−1,1,0,0,0 },    {−9,−112,2,41,9,4,1,2,4,39,−13,−16,−1,−1,−1,−1 },    {36,−8,110,−10,−34,−4,−3,−2,−15,−3,−29,20,10,1,2,0 },    {−7,−39,−7,−110,0,31,10,4,5,12,10,24,−23,−11,−2,−1 },    {−17,−4,32,−3,91,−2,−18,−5,−67,−20,24,−3,−21,20,7,−1 },    {−44,−7,−9,10,−67,8,15,1,−86,−1,40,7,14,−16,−3,−3 },    {−2,35,24,−8,0,−9,−5,−4,3,104,39,−43,−16,2,3,−3 },    {6,18,5,25,4,113,9,−17,−6,3,−22,−17,−6,3,−22,−17,−20,−20,23,7 },    {12,−9,16,32,−5,16,−28,−7,41,−10,86,60,−27,−12,3,−3 },    {13,−8,22,2,0,−8,89,12,17,−39,39,−53,−30,−6,−16,16 },    {1,−14,−1,−14,−43,3,−66,1,6,−38,10,−65,−39,44,21,−11 },    {3,−10,7,−15,9,28,3,−49,17,−8,38,−22,94,41,−8,2 },    {2,6,5,5,6,35,−5,102,2,4,9,1,23,31,−44,−32 },    {−3,3,−10,9,−13,3,47,−11,−2,17,−10,41,−28,97,34,−15 },    {0,4,−2,6,−6,−52,−6,2,−15,3,−34,14,−105,−28 },    {3,−1,3,−2,7,−7,16,−14,4,−8,2,−9,6,−29,29,−118 },   },   {    {103,7,−63,−5,4,0,−2,0,−35,6,21,−3,0,0,0,0 },    {2,−100,−12,68,10,−2,−1,1,−2,29,−14,−22,5,1,−1,−1 },    {56,−8,89,9,−60,−5,−2,−1,−16,2,−21,21,18,−3,0,−1 },    {−11,58,5,87,10,−49,−11,−3,−31,−17,21,−12,25,15,−4,−2 },    {31,34,−15,15,−1,−18,1,3,90,22,−63,−29,13,4,−1,3 },    {28,−11,38,−5,95,0,−41,0,5,−37,−22,26,−9,24,5,−3 },    {10,15,44,−10,18,22,−13,−3,3,66,47,−63,−49,22,12,−3 },    {−1,30,2,32,8,100,7,−29,−26,−5,−35,−23,4,−8,30,7 },    {11,7,−1,46,−6,24,−13,−9,53,5,49,64,−44,−41,11,3 },    {−18,8,−17,−11,−8,9,−88,−8,−13,62,−9,31,45,−1,2,−24 },    {−3,−10,−20,5,−53,0,−59,5,7,−54,−9,−22,−50,46,31,−18 },    {−8,13,−5,3,7,−35,−6,37,−42,22,−56,8,−58,−58,32,14 },    {2,−13,4,−17,0,−30,−31,−82,6,−24,10,−35,14,−57,19,40 },    {−5,0,−10,−2,0,−28,41,−67,−8,29,−18,45,−15,54,52,−8 },    {2,−7,3,−7,4,−1,5,51,15,−4,29,−4,50,1,97,26 },    {−5,2,−6,3,−10,8,−17,9,−5,14,−4,18,−9,41,−26,113 },   }  },  { //3   {   { −87,77,−22,6,−4,3,−2,1,38,−28,2,2,−1,1,−1,1 },    {−54,−29,89,−48,17,−8,5,−2,14,23,−35,9,2,−3,2,−1 },    {51,70,5,−76,32,−11,2,−3,−21,−19,−15,28,−2,−3,2,0 },    {−30,−45,−67,−28,78,−24,8,−1,6,14,14,25,−20,−2,3,−2 },    {0,−38,−27,−64,−41,66,−17,5,36,−32,12,14,28,−14,−5,3 },    {−38,16,−26,−22,−25,31,−3,1,−85,65,−2,−4,18,−6,−2,1 },    {8,14,19,32,64,66,−67,13,3,10,−4,−3,−2,−38,7,6 },    {18,6,−34,16,−17,4,3,−3,42,47,−95,35,0,−1,−4,3 },    {−3,−6,−12,−18,−25,−62,−77,58,1,1,−10,−18,7,−6,40,−3 },    {−17,−17,9,22,−22,6,−24,2,−46,−47,−13,84,−50,13,2,4 },    {12,15,7,−21,−19,17,−20,5,34,55,42,−1,−81,44,−8,3 },    {−5,−8,−10,−12,−4,−18,−55,−95,−8,−14,−20,−35,−9,9,−23,33 },    {5,9,9,7,−24,−27,−2,−46,20,32,45,47,7,−77,31,−3 },    {3,5,10,10,7,−24,−25,19,10,20,29,44,56,25,−78,36 },    {1,3,3,7,10,8,−25,−35,6,9,11,24,44,65,37,−77 },    {1,1,3,4,8,12,14,−4,3,5,8,12,23,40,75,89 },   },   {    {−112,22,27,1,0,1,1,0,50,−7,−12,−1,−1,0,0,0 },    {−15,−112,22,32,8,0,0,1,8,44,−5,−12,−3,−1,0,−1 },    {25,11,113,−15,−26,−7,0,0,−17,−4,−38,1,9,3,1,0 },    {7,34,10,114,5,−22,−9,−1,−8,−9,2,−35,−7,7,2,1 },    {48,−11,−12,7,−25,−5,−4,1,104,−29,−37,2,4,3,−1,1 },    {−17,−7,−21,8,−117,−6,20,6,−18,6,15,−9,27,10,−4,−1 },    {−2,−21,14,27,−4,91,10,−11,−9,−68,22,26,7,−19,−12,3 },    {12,36,−2,4,−4,77,14,−9,18,82,−18,−33,4,−16,−8,0 },    {−13,−5,−23,1,−14,15,−99,−27,−26,−5,−57,2,19,−9,16,13 },    {−11,−5,−29,13,19,−1,64,16,−30,−12,−89,15,32,7,−11,−7 },    {4,14,5,32,−6,−13,−10,13,9,43,9,107,15,−31,−6,−3 },    {0,−1,−1,2,−3,−21,30,−120,1,4,−2,12,−9,−5,−13,15 },    {−3,−1,−8,3,−28,7,12,11,−12,−2,−31,15,−112,−21,28,7 },    {−1,−3,−1,−6,−1,−26,6,12,−2,−11,0,−33,13,−114,−26,21 },    {1,0,2,3,5,3,24,−2,4,1,9,1,31,−13,111,47 },    {0,0,−1,0,−1,−2,1,−18,1,−4,1,−7,7,−29,43,−115 },   }  } };

TABLE 10 const int8_t g_lfnst_2×8_8×2[4][2][16][16] = {  { //0   {   { 101,−15,−70,13,14,−6,−2,2,−20,3,11,−1,−1,−1,0,2 },   { 6,96,−28,74,8,15,−3,−1,8,−19,0,13,−2,−4,3,0 },   { 52,−10,72,−45,53,23,8,−7,−50,11,17,−2,−1,4,0,−1 },   { −43,9,−22,7,23,−2,−3,−4,−81,5,77,−15,−25,8,5,−2 },   { 9,66,3,68,−65,−40,29,0,−3,−8,8,−16,7,11,−4,−1 },   { 30,29,62,14,69,−61,−38,19,−9,−1,−5,7,−8,4,7,−3 },   { 0,13,−4,−19,12,−4,6,1,13,86,−13,−84,8,27,−9,−4 },   { 18,25,20,47,10,80,−46,−27,37,4,30,−13,−31,−2,9,6 },   { 8,−20,3,−26,−15,−34,26,21,61,−7,52,−5,−73,17,22,−10 },   { 9,9,17,11,46,7,91,−63,2,20,−2,23,−10,−18,5,2 },   { 7,−11,9,−7,24,7,16,−26,7,−78,7,−51,26,63,−30,−14 },   { 5,0,12,−4,12,0,13,32,33,8,68,4,60,−37,−52,37 },   { 3,10,7,20,21,52,47,94,−13,−1,−15,9,−6,24,6,−26 },   { 1,−2,5−2,5,3,17,18,−14,−39,−23,−63,−26,−57,27,69 },   { 2,−3,2,−2,2,−1,2,−2,11,−11,23,−26,53,−42,77,−66 },   { 2,2,1,0,−1,−2,−2,0,−6,−11,−12,−23,−38,−63,−73,−70 },   },   {   { 104,29,−61,−27,2,3,−2,2,−14,−3,7,3,−1,−1,1,3 },   { 14,−92,−44,60,42,0,−12,1,−6,10,5,−3,−2,−3,−3,0 },   { −63,33,−78,−31,58,17,−4,−1,22,−5,3,3,−6,−6,1,0 },   { 17,−4,−8,3,−14,−5,7,9,98,35,−−60,−34,3,6,−1,1 },   { 6,66,6,78,32,−60,−31,4,5,−12,−4,−2,3,5,2,1 },   { −28,3,−62,33,−83,−31,−48,15,−9,−8,16,−5,3,1,−3,1 },   { −1,15,8,1,8,−5,12,−2,−11,82,55,−54,−52,−1,15,0 },   { 5,31,−1,59,−14,97,8,−43,12,−4,10,1,2,−5,−5,1 },   { −10,9,−9,0,11,8,14,−15,−65,23,−62,−54,46,41,0,−13 },   { 13,8,20,11,50,−1,106,27,3,5,−2,26,11,−12,−10,−1 },   { −6,8,−8,−1,−11,−13,−17,−25,−10,77,−11,57,44,−54,−29,1 },   { 4,−11,−1,−13,8,−42,29,−112,15,−18,6,−2,−12,10,−2,−10 },   { −3,2,−3,7,−4,0,8,−10,−30,−7,−63,5,−60,−56,48,45 },   { −2,2,−7,7,−6,5,−1,3,−2,30,−20,69,−35,67,47,−47 },   { −1,−4,0,−2,3,−2,3,−8,13,12,27,10,55,28,72,79 },   { −2,2,−1,0,0,0,−1,−1,−8,11,−12,21,−37,55,−74,75 },   }  },  { //1  {    { 117,−22,−24,−1,−8,0,−3,0,−38,8,8,0,2,0,0,0 },   { −18,−102,58,−6,14,1,5,−1,−33,30,−3,−1,−1,−1,0,−1 },   { 25,−45,−12,25,−6,11,−4,3,109,10,−29,−5,−8,−1,−3,0 },   { −35,−37,−87,64,7,9,3,2,−24,7,35,−14,1,−2,−1,−1 },   { 5,14,53,84,−73,6,−16,−1,−14,−11,−3,−19,12,3,2,1 },   { −11,−33,−17,−49,−69,53,15,5,2,−63,18,16,13,−2,−5,2 },   { 11,−18,17,28,55,−29,−8,−3,1,−104,2,4,−5,9,3,3 },   { −11,−6,−29,6,−5,8,−9,2,−33,−9,−115,24,10,4,6,2 },   { 5,14,16,12,43,100,−57,4,−5,4,7,11,−11,−17,3,2 },   { −9,−14,−13,−22,−24,−43,−101,48,2,2,12,7,−4,10,15,−3 },   { 0,1,5,25,−10,−12,10,0,2,11,18,116,−37,−2,−13,−2 },   { 4,8,9,11,12,13,43,117,−3,0,−3,−3,2,7,−5,−9 },   { 2,0,3,7,15,−9,−5,2,12,6,10,33,115,−39,2,−3 },   { 0,−1,−1,−1,−10,−16,6,8,−3,−10,−6,−11,−35,−116,29,−13 },   { 1,0,1,2,0,9,16,−5,5,4,7,11,3,29,117,−34 },   { 0,0,0,0,0,1,−7,−9,−1,−4,−3,−4,0,5,−35,−122 },   },   {   {−124,12,18,5,4,2,2,1,21,−5,−3,−1,−1,0,0,0 },   {13,116,−27,−23,0,−4,−4,−1,37,−11,−5,2,−1,−1,0,0 },   {18,−38,3,11,0,2,0,2,118,−2,−25,−4,−5,−2,−3,0 },   {18,27,117,−1,−26,−1,−3,−6,2,31,−7,−3,0,1,0,−1 },   {−7,7,−32,15,8,−1,−1,1,2,120,−10,−20,−2,−5,−1,−2 },   {6,22,−2,123,−5,−14,−7,−3,−5,−17,2,0,0,0,−1,0 },   {4,2,12,2,40,3,−4,−2,24,9,117,−4,−14,−5,−5,−3 },   {−6,−4,−22,0,−116,25,11,3,8,4,39,−8,−3,2,0,−3 },   {4,6,2,12,18,122,−24,−15,−3,−1,−10,9,−1,−4,−1,2 },   −2,−2,−2,3,−7,−8,4,−3,5,20,3,125,−1,−9,−6,−2 },   {5,7,4,8,14,19,124,−16,0,0,−2,−3,2,−1,0,0 },   {−3,−4,−6,−5,−5,−15,−11,−121,−1,−2,−4,−4,−36,3,0,−6 },   {1,2,2,1,0,3,6,36,−7,−2,−12,1,−121,8,9,2 },   {0,−1,1,−1,−4,−3,0,0,−3,−6,−4,−9,−8,−127,8,−1 },    {0,−1,0,1,2,0,−1,−3,4,3,5,6,8,6,127,5 },   {1,1,0,0,2,2,0,6,−1,−3,−3,−1,1,1,5,−128 },   }  },  { //2   {   { −111,12,42,0,0,2,1,0,42,−10,−15,2,−1,0,0,0 },   { −9,−112,5,42,9,1,0,2,5,38,−14,−16,−1,0,−1,−1 },   { −39,6,−110,9,33,5,4,2,16,5,28,−21,−10,−2,−2,0 },   { 5,41,6,109,2,−33,−10,−3,−6,−10,−10,−24,25,12,1,0 },   { 1,−1,31,−8,110,−2,−24,−5,−28,−19,8,−8,−26,26,9,0 },   { −44,−9,6,7,−27,1,9,−1,−102,−4,53,7,3,−6,−1,−3 },   { −1,−34,−25,10,2,7,7,2,−8,−103,−36,45,18,−4,−6,4 },   { −2,−17,−3,−26,1,−111,−5,17,15,−5,29,16,18,19,−28,−7 },   { 13,−8,15,35,−6,21,−28,−7,45,−11,83,58,−31,−11,4,−3 },   { −14,7,−21,−4,2,6,−85,−8,−18,40,−32,57,39,9,11,−20 },   { 0,−13,−5,−12,−47,0,−68,6,2,−38,2,−59,−41,45,20,−13 },   { 4,−11,8,−16,5,33,3,−42,19,−13,42,−30,90,42,−9,−2 },   { −2,−6,−4,−8,−3,−36,−9,−98,0,−7,−4,−8,−9,−46,47,27 },   { −3,2,−11,7,−12,−5,49,−26,−2,16,−12,41,−25,90,45,−6 },   { 0,−4,2,−6,5,−7,2,62,6,−4,16,−4,36,−15,99,29 },   { −3,1,−3,0,−7,5,−19,6,−4,8,−1,9,−5,26,−30,119 },  },  {   { 106,6,−57,−5,2,0,−2,0,−37,8,20,−3,0,0,1,0 },   { 2,−105,−11,59,8,3,0,1,−2,31,−17,−21,4,1,0,−1 },   { 51,−8,91,7,−60,−3,0,−1,−14,2,−25,23,18,−5,1,0 },   { −10,52,2,95,0,−45,−7,−3,−28,−9,15,−20,28,11,−3,−1 },   { 30,31,−17,12,−7,−12,1,2,95,34,−54,−27,11,1,−2,2 },   { 30,−6,34,7,92,−10,−34,0,10,−47,−31,25,−7,26,4,−1 },   { 9,13,51,−8,31,1,−18,−3,2,72,54,−45,−41,18,7,−4 },   { −5,26,4,4,9,78,6,−18,−44,0,−57,−55,9,10,26,5 },   { 6,15,−3,53,0,70,−13,−24,31,5,30,58,−26,−37,19,3 },   { −17,12,−8,−17,28,−5,−36,0,−29,74,−24,49,60,−23,−8,−12 },   { −11,−4,−22,−2,−47,−3,−98,4,−2,−7,−12,−1,−27,39,26,−28 },   { −5,12,−4,9,4,−39,9,31,−30,15,−47,9,−77,−56,29,16 },   { −4,11,−8,15,−3,12,52,35,−8,28,−16,47,−26,81,−3,−35 },   { −3,−7,−7,−10,−2,−38,23,−99,−2,11,−7,18,−5,24,57,10 },   { 2,−6,3,−5,6,−1,10,57,14,−5,24,−3,42,−1,102,3 },   { −5,1,−5,2,−9,6,−16,15,−3,13,0,16,1,39,−8,117 },   }  },  { //3   {   { 97,−67,11,−3,3,−3,1,−1,−42,24,2,−1,1,0,1,−1 },   { 53,60,−83,29,−14,8,−5,1,−16,−31,30,−2,0,2,−2,2 },   { 38,73,50,−70,16,−8,2,−3,−20,−19,−25,21,3,−3,3,0 },   { 17,29,69,73,−60,4,−5,1,−4,−9,−12,−33,9,8,−1,2 },   { 3,21,13,59,91,−50,2,−4,−10,12,−7,−5,−33,1,7,−2 },   { −42,21,−9,−7,−17,−6,12,−1,−99,57,13,−13,9,4,0,−1 },   {  3,−6,−18,−15,−46,−101,41,−3,16,−3,−12,1,−9,35,6,−5 },   { 19,25,−31,4,−7,12,−10,4,38,82,−76,−7,9,−8,1,3 },   { −7,−8,−6,−11,−11,−40,−106,36,−13,−12,−8,−11,2,−13,33,8 },   { 15,17,13,−31,13,−8,2,12,39,37,69,−81,10,−1,5,−5 },   { −7,−14,−17,−3,29,−7,19,−5,−18,−45,−48,−62,78,−7,−4,0 },   { −2,−2,−4,−11,−6,−5,−36,−112,−1,−2,−4,−25,−23,4,−23,25 },   { 3,7,9,18,−4,−35,−2,−30,14,21,35,50,71,−67,−1,1 },   { −2,−3,−6,−5,−17,4,33,−12,−6,−14,−14,−27,−47,−83,66,−12 },   { 1,2,1,5,7,17,−9,−37,5,6,6,13,28,52,82,−64 },   { 1,0,2,3,6,9,17,1,3,4,7,8,15,25,60,107 },   },   {   { −110,37,23,0,0,1,1,0,46,−12,−11,0,0,0,−1,0 },   { 25,105,−50,−19,−6,−3,−1,0,−8,−40,12,10,1,1,0,1 },   { 28,31,102,−43,−27,−3,2,−1,−19,−8,−34,7,11,1,1,0 },   { 14,30,31,111,−10,−20,−7,−1,−8,−10,−2,−33,−6,8,2,1 },   { −41,18,9,−4,36,0,1,0,−103,35,32,−5,−6,−3,3,−1 },   { 21,10,20,−1,114,−10,−16,−4,26,−10,−21,10,−25,−10,5,1 },   { −6,−25,21,17,5,66,−3,−6,−18,−86,39,33,3,3,−13,−8,2 },   { 12,29,−2,13,4,−101,−1,−14,14,57,−22,−18,9,−22,−10,3 },   { −16,−12,−30,14,−6,−1,−56,−6,−39,−18,−90,30,27,−8,10,8 },   { 2,0,10,−19,−15,0,−109,−8,13,6,43,−31,−17,−10,16,12 },   { 4,13,10,25,−18,−16,−11,−9,13−,43,24,103,−18,−34,−2,0 },   { −2,−4,−3,−5,−5,−12,12,−124,−3,−6,−3,−8,−14,4,−13,12 },   { −3,−3,−6,−5,−23,19,12,12,−12,−9,−29,−1,−114,6,32,2 },   { −1,−3,−3,−4,−7,−21,21,7,−4,−13,−5,−31,−1,−115,−1,31 },   { −1,−1,−2,−4,−4,−7,−18,15,−6,−2,−11,−5,−31,0,−120,−12 },   { 0,−1,−2,−1,−3,−3,−6,−13,−1,−5,−3,−9,−1,−31,13,−122 },   }  } };

TABLE 11 const int8_t g_lfnst_2×8_8×2[ 4 ][ 2 ][ 16 ][ 16 ] = {  {//0  {    {103,−12,−70,9,13,−6,−2,2,−21,3,11,0,−1,−1,0,2 },   {7,96,−18,−77,−3,16,0,−2,2,−20,3,14,−1,−3,3,0 },   {49,−21,78,−37,51,20,8,−6,−51,11,16,−2,0,4,0,−2 },   {44,−7,24,−8,−20,0,5,82,−2,−76,11,23,−6,−4,2 },   {13,66,10,71,−59,−45,23,2,−5,−7,7,−18,8,13,−4,−1 },   {27,25,60,10,77,−53,−42,17,−11,−8,−3,13,−11,2,9,−3 },   {−4,−19,−2,16,−17,−2,0,0,−20,−89,0,81,11,−23,1,3 },   {19,19,20,45,4,77,−34,−26,41,−9,41,2,−41,−9,13,6 },   {5,−24,1,−30,−18,−49,30,26,52,−15,51,6,−67,11,20,−9 },   {−8,−10,−16,−11,−42,−3,−95,51,0,−25,2,−32,7,24,−2,−3 },   {−8,10,−10,7,−28,−10,−26,31,−9,71,−6,54,−20,−64,25,16 },   {−5,0,−12,3,−12,−3,−10,−40,−31,−10,−67,−6,−66,25,57,−24 },   {2,10,5,19,17,50,42,95,−16,−3,−21,6,−13,24,11,−27 },   {1,−2,6,−3,7,1,18,20,−12,−38,−16,−61,−18,−62,18,74 },   {3,−3,2,−2,3,−2,3,−2,10,−10,22,−27,52,−43,79,−65 },   {2,2,1,0,−2,−2,−4,−1,−7,−12,−13,−23,−38,−62,−72,−71 },   },   {   { 106,23,−62,−21,4,2,−3,2,−15,−1,7,2,−1,−1,1,3 },   { −11,97,35,−63,−37,1,12,0,7,−11,−5,3,1,3,3,0 },   { 57,−28,79,28,−67,−12,9,0,−23,5,−2,−3,7,6,−1,0 },   { 2,61,14,73,37,−55,−31,1,−30,−23,16,8,2,3,3,0 },   { 16,16,−9,33,−12,−30,0,12,94,27,−55,−29,5,6,−1,2 },   { 33,−8,65,−35,78,21,−47,−8,17,11,−24,5,−2,−1,4,−1 },   { −2,14,8,−3,13,−9,14,0,−12,88,50,−56,−44,−1,12,−2 },   { 6,32,2,55,−8,99,−2,−44,16,1,12,−4,−1,−4,−3,2 },   { −9,10,−10,1,13,8,17,−16,−61,24,−70,−46,54,30,−5,−10 },   { 14,7,20,13,48,1,109,15,9,−2,1,30,6,−11,−8,0 },   { −6,9,−7,2,−12,−11,−17,−18,−7,72,−7,66,40,−59,−24,5 },   { 3,−11,−3,−14,3,−42,17,−115,14,−13,4,−1,−12,7,−2,−9 },   { 3,−2,4,−8,5,−2,−7,6,29,6,61,−3,68,53,−50,−40 },   { 2,−3,8,−8,7,−6,1,−3,4,−29,21,−65,35,−69,−38,57 },   { −2,−4,0,−2,2,−2,3,−8,12,12,24,13,49,34,79,74 },   { 2,−2,1,0,0,0,2,8,−12,13,−21,37,−55,73,−76 },   }  },  { //1   {   { 118,−19,−24,−1,−8,0,−3,0,−36,7,7,0,2,0,0,0 },   { −17,−105,52,−5,14,2,6,−1,−36,29,0,−1,0,−2,0,−1 },   { 24,−46,−13,16,−2,10,−2,3,110,10,−27,−4,−8,−1,−3,0 },   { −33,−35,−100,44,19,9,6,3,−19,4,33,−9,−1,−2,−2,−1 },   { 2,0,−29,−88,82,−13,10,0,9,10,−6,20,−12,−4,0,−1 },   { −12,−25,−23,−69,−74,49,19,5,2,−35,18,16,12,−4,−5,1 },   { 7,−25,12,18,34,−11,−6,−3,−1,−117,−3,8,1,8,3,3 },   { −11,−5,−31,3,−13,2,−3,2,−32,−1,−117,15,13,5,5,3 },   { −4,−13,−13,−14,−37,−108,44,5,7,−4,3,−16,9,17,−1,−3 },   { −9,−12,−11,−20,−18,−32,−111,38,0,2,10,5,−3,6,15,−1 },   { 0,0,−2,24,11,16,−9,−5,−1,−9,−12,−119,28,1,11,2 },   { 4,6,9,8,13,14,32,121,−3,−1,−2,−5,6,8,−2,−7 },   { −2,0,−2,−6,−15,6,5,3,−11,−7,−10,−26,−119,29,−3,4 },   { 0,−1,0,2,−7,−15,5,−10,−3,−9,−6,−7,−26,−120,26,−8 },   { −1,0,0,−2,0,−8,−15,5,−4,−3,−6,−10,0,−26,−122,20 },   { 0,0,0,0,0,−3,5,7,1,4,4,3,2,−4,21,126 },  },  {   { 124,−15,−16,−5,−4,−2,−2,−1,−22,6,4,1,1,0,0,0 },   { −17,−114,26,22,0,5,4,1,−42,−10,6,−2,1,1,0,0 },   { 18,−42,7,12,0,2,0,1,116,0,−24,−4,−5,−1,−3,0 },   { −2,16,6,17,−1,−2,−2,−1,0,124,−11,−20,−2,−5,−1,−3 },   { −18,−25,−121,−6,25,2,3,5,1,10,1,−2,0,−2,0,0 },   { −5,−19,15,−123,7,8,9,3,4,19,−5,3,−1,0,0,−1 },   { 3,1,11,1,47,−4,−3,−3,23,8,115,−4,−14,−5,−4,−2 },   { 6,4,20,6,113,−28,−10,−3,−10,−5,−46,5,0,−2,1,2 },   { 3,5,3,6,21,118,−39,−13,−2,1,−7,9,−4,−4,0,1 },   { −3,−4,−2,−12,−7,−17,−84,9,−3,−14,−1,−92,−3,5,4,1 },   { −4,−7,−3,−4,−14,−31,−86,−12,3,14,4,85,−2,−1,−3,−2 },   { 3,4,6,5,8,15,7,120,2,1,5,−1,37,−1,3,6 },   { 0,2,0,2,−3,2,8,37,−7,−2,−11,1,−121,9,6,1 },   { 0,1,−1,1,5,3,1,−3,3,6,4,3,8,127,−6,5 },   { 0,−1,0,1,0,−1,0,−5,4,3,4,4,5,6,127,−2 },   { −1,−1,0,0,−2,−2,−1,−6,1,3,2,2,−2,−5,2,127 },   }  },  { //2   {   {−111,11,41,0,0,2,1,0,43,−10,15,2,−2,0,0,0 },   {9,112,−4,−41,−9,−1,0,−2,−6,−39,13,16,1,0,1,1 },   {39,−7,110,−9,−30,−5,−4,−2,−15,−5,−29,21,10,3,2,0 },   {6,40,7,109,4,−33,−10,−3,−6,−12,−11,−25,25,12,2,1 },   {1,−1,29,−8,110,2,−22,−5,−28,−20,9,−9,−27,25,9,1},   {−45,−8,8,7,−28,−1,7,−2,−102,0,53,4,3,−5,0,−4 },   {−3,−32,−25,13,0,24,10,0,−14,−99,−40,43,18,−7,−2,4 },   {2,22,4,21,−3,106,9,−14,−16,20,−29,−30,−17,−19,28,7 },   {13,−7,15,37,−9,31,−30,−8,42,−14,81,52,−35,−9,9,−4 },   {−13,6,−22,−5,−3,5,−93,−8,−18,35,−35,48,27,14,19,−20 },   {−1,15,3,15,45,3,53,−3,−4,45,−2,70,41,−44,−20,9 },   {−3,9,−7,16,−6,−34,−2,35,−18,8,−40,23,−94,−47,8,3 },   {−1,−7,−3,−8,−2,−34,−1,−102,2,−6,−2,−6,−5,−39,48,30 },   {3,−3,11,−9,12,3,−52,25,3,−17,14,−40,32,−92,−33,11 },   {0,4,−1,5,−5,10,−10,−61,−6,3,−14,0,−31,9,−103,−22 },   {−4,1,−3,1,−8,5,−18,11,−5,8,−1,10,−4,27,−25,120 },   },   {   { 107,4,−54,−4,1,0,−3,0,−39,9,20,−3,0,0,1,0 },   { 0,−107,−10,54,8,6,1,1,−2,32,−18,20,3,0,0,−1 } ,   { −47,7,−92,−9,62,4,−1,1,15,−3,26,−22,−20,5,0,0 },   { −34,−6,14,29,7,−6,−2,−3,−100,−36,51,14,3,3,1,−1 },   { 1,−57,8,92,12,43,2,2,−14,−21,3,36,−26,−8,3,1 },   { −29,0,−25,−25,−82,16,29,1,−12,59,41,−24,−1,−23,−4,−1 },   { −12,−9,−59,14,−52,−2,18,2,3,−67,−45,34,38,−17,−8,2 },   { −7,25,2,−7,8,58,12,−10,−48,1,−68,−66,11,12,21,5 },   { −4,−20,1,−51,−2,−87,12,25,−19,−8,−18,−48,18,34,−24,−2 },   { 18,−12,13,17,−26,12,35,0,30,−67,32,−53,−59,18,10,12 },   { −10,−3,−17,−2,−44,−1,−100,3,−2,−6,−8,−6,−28,40,28,−27 },   { −5,12,−6,11,−1,−39,10,33,−25,15,−45,16,−81,−54,24,11 },   { −3,9,−8,14,−5,14,51,35,−6,23,−13,45,−22,87,1,−34 },   { 4,6,7,9,3,35,−24,99,2,−10,7,−17,7,−23,−58,−14 },   { −1,6,−3,6,−6,4,−7,−58,−11,4,−21,1,−43,4,−100,16 },   { 4,−2,5,−3,9,−6,17,−12,3,−11,2,−15,5,−36,18,−117 },   }  },  { //3  {    { 103,−58,4,−2,3,−2,1,−1,−45,20,4,−1,1,0,1,−1 },   { −46,−76,75,−19,12,−8,4,−1,16,36,−26,−1,0,−2,2,−2 },   { 30,67,67,−66,14,−9,2,−3,−17,−17,−30,19,3,−2,3,0 },   { −15,−25,−65,−86,48,1,6,−1,3,8,12,34,−5,−8,0,−2 },   { −2,−17,−4,−49,−103,41,0,4,6,−13,6,3,33,2,−6,1 },   { −44,20,−3,−5,−14,−11,14,−1,−102,51,19,−14,6,5,0,−1 },   { 6,−1,−20,−11,−40,−107,28,1,20,6,−19,−2,−9,33,9,−4 },   { 18,30,−26,−2,−5,23,−11,4,33,93,−59,−19,10,−11,1,3 },   { −9,−10,−9,−7,−6,−28,111,25,−20,−15,−22,−7,5,−15,29,10 },   { −12,−15,−22,30,−6,13,18,−14,−35,−29,−87,64,6,2,−7,2 },   { −4,−12,−16,−17,27,2,21,−15,−12,−41,−39,−92,50,8,−3,2 },   { 1,0,1,8,11,2,29,116,−1,−4,−3,9,31,−3,20,−22 },   { 2,4,5,16,10,−34,−3,−28,11,12,24,34,98,−47,−11,0 },   { −1,−2,−4,−2,−18,−7,32,−8,−3,−11,−8,−17,−36,−103,50,−5 },   { 1,2,1,4,5,18,0,−33,4,5,7,11,20,41,99,−54 },   { 1,0,2,2,4,6,18,4,3,3,7,8,12,17,51,113 },   },   {   { −110,39,20,0,0,1,1,0,46,−13,−10,0,0,0,−1,0 },   { −26,−100,59,9,6,4,2,0,5,41,−15,−8,0,−2,0,−1 },   { 28,41,94,−48,−31,−3,3,−1,−19,−12,−33,8,13,2,1,0 },   { −35,29,20,32,34,−8,0,−1,−100,28,26,−14,−8,0,3,−1 },   { 29,19,35,105,−6,−17,−7,0,29,−21,−17,−28,−7,7,0,1 },   { 19,8,18,−17,113,−13,−14,−3,24,−12,−20,17,−25,−10,5,1 },   { −6,−27,20,13,6,67,−5,−4,−20,−87,37,29,4,−14,−8,2 },   { 12,29,−2,10,10,100,−6,−13,14,57,−22,−16,8,−24,−10,3 },   { 16,12,31,−17,7,3,49,2,36,16,91,−39,−26,9,−7,−6 },   { 1,1,10,−17,−14,−4,−113,1,10,6,37,−28,−14,−9,19,11 },   { −5,−12,−14,−25,19,13,6,5,−14,−40,−32,−103,20,31,1,0 },   { −2,−4,−2,−3,−6,−13,1,−125,−2,−6,0,−7,−10,−1,−15,14 },   { −3,−3,−6,−6,−23,20,11,6,−12,−8,−28,−2,−114,14,31,2 },   { −1,−3,−3,−4,−9,−20,21,9,−4,−12,−6,−30,−6,−115,13,27 },   { −1,−1,−1,−5,−5,−10,−16,18,−6,−2,−9,−6,−32,−10,−120,−5 },   { 0,0,2,1,4,2,8,13,1,5,3,7,3,27,−8,124 },   }  } };

Each of Table 12 to Table 14 shows an example in which LFNST kernels areconfigured of a total of 2 LFNST sets, wherein each LFNST set isconfigured of two LFNST kernel candidates, i.e., an LFNST kernel matrix.The LFNST kernels that are proposed in Table 12 to Table 14 are definedaccording to the grammar of C/C++ programming languages. And, ing_lfnst_2×8_8×2[2][2][16][16], which is an array that stores LFNSTkernel data, [2] indicates that the kernels are configured of a total of2 LFNST sets, [2] indicates that each LFNST set is configured of 2 LFNSTkernel candidates, and [16][16] indicates that each LFNST kernel isconfigured of a 16×16 matrix. Each 16×16 matrix shown in Table 12 toTable 14 represents a matrix that is used in forward LFNST transform.That is, one row becomes one transform basis vector (1×16 vector), whichis then multiplied by input data that is configured of primary transformcoefficients.

TABLE 12 const int8_t g_lfnst_2×8_8×2[2][2][16][16] = {  { //0   {   { 99,−25,−69,23,13,−7,−3,2,−18,5,10,−3,−1,0,0,2 },   { 6,−86,55,56,−45,−1,10,−1,−15,16,1,−12,5,4,−4,0 },   { 55,21,53,−70,−38,35,1,−8,−44,9,20,−3,−4,2,2,0 },   { 41,−2,17,−12,−24,12,0,3,80,−16,−74,26,23,−12,−4,3 },   { −9,−75,6,−53,72,28,−39,6,1,8,−9,12,−6,−7,4,0 },   { 34,24,64,13,53,−80,−15,27,−6,5,−8,2,−4,6,4,−4 },   { −7,6,−13,−19,7,−7,12,0,−5,76,−39,−70,50,14,−23,1 },   { 11,37,22,57,18,65,−76,−7,10,10,7,−22,5,3,−4,6 },   { −10,10,−2,18,10,5,−1,−17,−73,−10,−47,49,54,−47,−4,16 },   { 14,8,23,18,51,22,66,−83,17,17,5,5,−17,−3,6,−1 },   { −5,10,−8,6,−18,−2,−8,16,7,87,−7,39,−46,−45,46,−3 },   { 4,−2,9,−5,6,−5,11,17,35,12,68,5,41,−59,−35,58 },   { −4,−10,−9,−19,−29,−54,−62,−87,8,−1,4,−11,−4,−14,2,15 },   { 1,−1,3,0,5,5,13,12,−10,−37,−30,−71,−35,−51,47,50 },   { 3,−3,2,−2,2,−2,0,−3,14,−14,30,−21,58,−35,67,−75 },   { 2,2,1,1,0,0,−1,1,−5,−8,−10,−21,−40,−65,−76,−65 },   },   {   { 100,43,−51,−38,−6,2,1,1,−15,−5,6,5,0,−1,2,3 },   { 24,−87,−61,42,49,8,−9,1,−9,9,9,−1,−5,−5,−2,1 },   { 68,−35,79,42,−34,−28,−7.1,−23.5,−4,−1,6,6,0,−1 },   { 22,−3,−4,4,−5,−4,4,7,93,48,−51,−45,8,7,3,0 },   { −1,67,−10,79,43,−46,35,−1,6,−13,0,−1,1,2,3,2 },   { 18,15,54,−18,75,61,−25,−29,1,28,17,−6,−21,−9,4,1 },   { −9,12,−10,12,−26,−19,18,9,−16,66,73,−34,−55,−16,11,3 },   { −8,−21,9,−67,37,−81,−48,20,−14,11,−10,−11,−10,5,10,1 },   { −11,12,−11,7,5,16,14,−10,−70,21,−47,−69,17,48,17,−9 },   { 8,9,20,0,53,−17,92,62,−5,7,−5,16,11,−4,−6,−2 },   { 7,−10,9,−1,5,10,9,24,18,−83,17,−46,−56,27,40,17 },    { 5,−6,2,−10,14,−25,24,−49,29,−10,56,−14,33,65,−26,−46 },   { 1,−4,−2,−4,−10,−38,50,−88,−7,−14,−32,3,−36,−39,10,17 },   { −2,−2,−3,3,−1,−1,6,−12,10,30,5,57,10,54,87,38 },   { 0,−3,4,−6,5,−5,4,−8,8,−7,33,−47,74,−34,9,78 },   { 2,−2,0,0,0,1,0,2,6,−8,8,−15,34,54,76,−78 },   }  },  { //1   {   { −93,71,−11,0,1,1,0,0,43,−27,−1,3,−1,0,−1,1 },   { 50,52,−88,35,−11,5,−5,1,−17,−27,32,−5,−1,2,−1,1 },   { −45,−69,−36,76,20,8,−3,3,23,22,17,−25,0,2,2,0 },   { 24,32,65,52,−77,19,−7,0,3,−19,−17,−21,18,2,−2,2 },   { −34,37,10,29,13,−22,7,−2,−91,60,−2,−19,0,4,3,−2 },   { 27,13,26,58,64,−66,10,−4,36,−15,−19,−10,−24,−15,3,−1 },   { −21,−22,34,−1,10,−12,7,−2,−41,73,82,−12,−7,8,−2,0 },   { 3,9,20,25,60,75,−70,13,−4,−2,3,−2,−9,−26,11,3 },   { −18,−22,−13,28,−8,−1,−5,1,−44,−49,−59,80,−17,3,2,1 },   { −5,−7,−8,−19,25,−60,−85,−58,−3,−5,−5,−14,7,6,28,−8 },   { −6,−14,−18,−6,27,0,12,−9,−18,−43,−50,−58,80,−21,1,4 },   { 4,7,11,21,9,−14,31,53,11,15,32,53,58,−62,13,−15 },   { −1,0,1,4,−10,−33,−45,−96,6,9,19,25,24,−43,13,18 },   { −1,−2,−5,−6,−14,1,28,−4,−4,−12,−14,−31,−57,−64,78,−20 },   { 1,2,1,4,8,16,1,−24,4,5,8,16,35,64,69,−70 },   { 1,1,1,2,4,7,14,12,3,5,6,9,15,32,65,102 },   },   {   { 111,−10,−44,−5,0,−1,−1,0,−42,4,16,2,1,0,0,0 },   { 2,−110,21,48,7,−1,0,1,2,35,−7,−14,−2,0,0,−1 },   { 40,6,104,−25,−46,−6,1,−1,−15,−4,−29,8,11,2,1,0 },   { 7,48,13,103,7,−41,−12,−2,−19,−9,4,−28,2,10,3,1 },   { 43,6,−16,−10,−3,−9,−5,2,105,−7,−54,−8,3,2,0,2 },   { −15,−9,−38,7,−110,2,39,7,4,8,11,−7,23,1,−8,−2 },   { −1,−32,−2,23,−3,4,−1,−2,0,−106,5,58,11,−3,−4,1 },   { 10,19,11,36,0,113,11,−30,3,5,−2,−5,1,−25,−5,5 },   { −17,0,−29,0,12,5,2,1,−51,−1,−101,−2,48,6,1,1 },   { 8,7,14,6,37,−11,116,20,2,−5,0,−2,−1,6,−22,−7 },   { 1,16,0,23,4,−16,−2,6,8,57,2,98,25,−39,−13,−4 },   { −1,−5,−2,−9,4,−30,17,−120,4,2,5,−1,16,1,−10,17 },   { 4,−1,11,−6,21,10,−18,13,17,0,41,−22,103,29,−34,−14 },   { 0,4,−2,9,−3,20,3,−13,2,18,−2,41,−16,108,24,−34 },   { 2,−1,3,1,7,0,21,6,8,2,16,1,38,−8,112,39 },   { 0,1,0,1,0,6,−3,17,−1,7,0,12,−7,38,−35,114 },   }  } };

TABLE 13 const int8_t g_lfnst_2×8_8×2[2][2][16][16] = {  {//0   {   { 99,−22,−70,21,13,−7,−3,2,−18,4,10,−3,−1,0,0,2 },   { 5,−87,51,61,−41,−4,10,−1,−15,16,1,−12,5,4,−4,0 },   { 55,17,55,−67,−43,35,3,−8,−44,9,20,−3,−4,2,2,−1 },   { 41,−4,17,−12,−25,11,1,3,80,−14,−74,24,24,−12,−5,3 },   { 8,74,−7,53,−71,−32,40,−5,−1,−7,8,−12,6,7,−4,0 },   { −35,−26,−65,−13,−54,77,20,−26,7,−4,7,−3,4,−6,−4,4 },   { 6,−6,11,19,−9,7,−10,−1,3,−78,35,74,−45,−17,22,0 },   { −11,38,−21,−58,−16,−68,73,14,−10,−9,−7,21,−5,−2,4,−6 },   { 10,−10,3,−16,−9,−4,0,16,73,7,49,−44,−59,−45,8,−15 },   { 14,7,22,16,50,18,70,−82,15,16,4,6,−17,−5,8,−1 },   { −4,10,−8,7,−18,−1,−9,17,6,86,−7,39,−45,−49,45,1 },   { 4,−2,9,−6,5,−8,8,12,36,11,68,3,42,−58,−36,58 },   { −5,−10,−10,−20,−30,−54,−61,−88,6,−1,1,−10,−6,−11,3,11 },   { 1,−1,3,0,4,4,12,12,−12,−39,−31,−70,−35,−49,44,54 },   { 3,−3,2,−2,2,−1,0,−2,13,−14,28,−23,56,−38,68,−74 },   { 2,2,1,1,0,0,0,2,−5,−8,−10,−21,−41,−64,−76,−64 },   },   {   { 102,41,−53,−35,−4,2,0,1,−15,−5,6,4,0,−1,2,3 },   { 22,−89,−58,45,47,6,−9,1,−9,10,9,−1,−4,−5,−3,0 },   { 66,−36,82,37,−37,−24,−5,0,−25,5,−4,1,6,6,0,−1 },   { 23,−2,−3,6,−6,−5,4,7,95,45,−53,−42,−6,7,2,1 },   { 2,65,−6,82,42,−47,−34,−1,4,−14,1,0,2,2,2,2 },   { −18,−14,−53,21,−80,−58,26,26,−1,−28,−14,5,19,9,−4,0 },   { −8,15,−9,15,−26,−13,21,6,−15,68,71,−38,54,−12,10,2 },   { 8,20,−7,63,−33,86,42,−24,17,−15,10,16,10,−7,−11,0 },   { −10,12,−12,10,4,20,13,−15,−67,20,−52,−67,21,47,14,−9 },   { 7,9,17,2,50,−16,96,58,−7,9,−6,16,15,−3,−8,−3 },   { 7,−10,10,−1,8,10,12,25,17,−81,17,−49,−55,31,37,15 },   { 5,−8,1,−12,16,−40,44,−91,22,−14,29,−8,10,−34,−18,−30 },   { 1,0,2,−3,−1,18,−29,49,21,7,56,−11,53,66,−24,−37 },   { −2,−2,−4,3,−1,0,6,−10,8,29,3,57,6,57,87,33 },   { 0,−3,4,−6,5,−5,4,−11,8,−6,32,−44,72,−32,12,82 },   { 2,−2,1,0,0,0,1,1,6,−9,9,−17,35,−55,77,−76 },   }  },  { //1   {   { −97,65,−6,0,−1,1,0,0,45,−25,−3,2,−1,0,−1,1 },   { 49,67,−82,24,−9,4,−4,1,−16,−31,29,−3,0,2,−1,1 },   { −39,−66,−58,72,−10,6,−2,3,19,22,23,−23,−2,1,−2,0 },   { −18,−25,−62,−73,69,−11,6,0,−1,14,16,25,−15,−4,1,−2 },   { 14,−31,−22,−52,−72,51,−3,4,51,−27,1,18,19,−9,5,1 },   { −44,11,−9,−26,−48,27,5,0,−89,42,22,−7,17,−5,−2,−1 },   { 17,31,−29,−4,−16,−12,5,1,34,91,−66,−6,8,−2,1,−2 },   { −4,−12,−9,−18,−49,−98,51,−3,2,−15,12,−2,1,29,−5,−4 },   { 16,19,25,−24,7,6,4,1,41,40,82,−67,−1,2,−4,−2 },   { 5,5,4,16,17,43,104,−46,4,2,0,14,0,−1,−26,2 },   { −4,−10,−16,−18,23,1,9,−7,−12,−36,−44,−85,66,−4,−1,3 },   { −2,−4,−6,−10,−13,−5,−41,−110,−2,−2,−8,−17,−35,15,−2,20 },   { 2,3,6,14,8,−30,−15,−38,11,13,30,43,82,−60,−1,5 },   { 0,−1,−3,−3,−14,−9,24,−2,−3,−11,−8,−25,−47,−93,63,−4 },   { 1,1,0,3,4,14,7,−21,5,37,12,26,49,94,−60 },   { 0,0,1,1,3,4,13,11,3,5,6,8,13,24,54,111 },      },   {   { 112,−22,−39,−2,0,−1,−2,0,−41,7,14,1,1,0,0,0 },   { −7,−109,36,42,3,0,0,0,5,33,−11,−13,−2,0,0,−1 },   { −37,−16,−102,33,44,5,−1,1,13,7,27,−9,−11,−3,−1,0 },   { −13,−45,−15,−105,−13,39,11,1,11,7,1,28,−2,−10,−3,−1 },   { −41,0,16,−4,3,4,5,−2,−107,15,51,2,−1,−2,0,−1 },   { −15,−13,−38,6,−112,9,35,5,3,2,9,−1,21,1,−8,−1 },   { −2,−30,5,23,3,9,−2,−3,−7,−107,14,55,6,−3,−3,1 },   { 11,20,13,33,3,114,4,−28,4,11,2,−3,−1,−23,−5,4 },   { 15,1,27,−3,−10,−9,3,−1,48,5,103,−3,−48,−4,−3,0 },   { 8,8,11,8,34,−8,119,9,2,−2,−3,0,7,4,−20,−6 },   { 1,13,0,23,0,−17,−5,9,9,54,2,102,16,−39,−10,−3 },   { −1,−4,−2,−8,3,−27,3,−116,7,4,11,−2,37,0,−17,11 },   { 4,0,10,−3,18,10,−20,38,17,1,39,−17,102,16,−33,−13   { 0,−3,1,−6,0,−18,1,13,−3,−17,−1,−38,10,−113,−11,33   { 1,−1,2,1,5,2,18,−1,9,3,16,4,37,−3,117,24 },   { 1,−1,0,−1,1,−4,0,−13,0,−7,−1,−10,2,−34,24,−120 },   }  } };

TABLE 14 const int8_t g_lfnst_2×8_8×2[2][2][16][16] = {  { //0   {   { 101,−17,−71,15,12,−5,−3,2,−18,4,10,−2,−1,−1,0,2 },   { 3,92,−34,−76,18,12,−3,−1,7,−16,2,12,−4,−3,4,0 },   { 54,−3,67,−48,−61,31,9,−6,−39,9,13,−2,1,2,0,−1 },   { 36,−4,12,−4,−24,4,3,5,86,−11,−76,17,23,−8,−5,3 },   { −4,−68,6,−61,67,41,−35,1,2,7,−10,16,−6,−10,4,1 },   { −37,−29,−66,−10,−58,68,31,−22,9,−5,8,−3,5,−5,−5,3 },   { 4,−5,10,23,−9,7,−14,−1,−6,−85,25,81,−22,−22,11,3 },   { −15,−37,−19,−50,−11,−74,60,24,−26,−12,−19,26,17,−5,−4,−4 },   { 8,−19,3,−22,−10,−20,17,18,65,−3,52,−21,−75,29,19,−11 },   { 13,7,21,14,47,13,83,−74,8,13,0,17,−14,−12,7,1 },   { −6.9,−8.7,−20,−5,−8,20,−1,77,−7,47,−3 7,−62,40,12 },   { 6,2,13,1,15,10,24,43,33,10,65,4,50,−40,−47,41 },   { 3,12,7,22,23,52,48,84,−18,−2,−24,9,−11,29,12,32 },   { 1,−2,4,0,6,4,15,15,−16,44,−28,−64,−27,−49,33,68 },   { 3,−4,3,−2,2,−2,2,−2,13,−15,26,−28,52,−43,73,−68 },   { −2,−2,−1,0,1,1,1,−1,6,11,14,25,42,64,74,64 },   },   {   { 107,32,−53,−27,−1,0,−2,2−16,3,6,3,0,0,2,3 },   { −20,99,42,−49,−40,−2,8,−1,9,−13,−5,2,3,4,3,0 },   { 57,−35,92,18,−47,−13,3,−2,−27,9,−4,−1,5,6,0,−1 },   { −23,0,−1,−12,8,7,−3,−7,−100,−41,52,32,1,−3,1,−1 },   { −9,−56,−92,−35,47,30,1,4,16,−4,−2,−4,−3,−1,−1 },   { 17,8,53,−33,92,41,−35,−16,4,21,−1,2,−9,−5,3,−1   { 6,−20,3,−15,9,−3,−20,2,14,−81,−65,43,47,6,−9,1 } },   { −10,−15,0,−52,21,−100,−23,29,−20,22,−15,−17,−4,7,8,−1 },   { 10,−14,12,−11,−7,−25,−17,16,61,−24,69,57,−33,−34,5,8 },   { −7,−7,−14,−5,−44,14,−105,−38,1,−8,4,−31,−16,12,10,1 },   { −8,9,−11,3,−16,−5,−24,−25,−11,72,−13,65,50,−39,−29,−5 },   { −4,7,1,10,−11,38,−35,110,−15,13,−13,5,0,−16,11,18 },   { −2,2,−2,6,−4,−5,−24,21,−6,−55,14,−70,−68,33,37 },   { −2,0−6,5,−3,3,1,−3,3,25,−10,63,−23,76,73,−5 },   { 1,4,−2,4,−4,4,−4,13,−10,−1,−26,18,−60,10,−53,−92 },   { −2,2,0,0,0,0,−1,−2,−6,8,−9,16,−34,53,−77,78 },   }  }, { //1   {   { 109,−38,−12,0,0,−1,0,0,−51,17,8,−2,1,0,1,0 },   { −29,−98,60,−1,7,−3,2,0,9,40,−21,−4,0,−1,1,−1 },   { −26,−47,−91,57,2,4,−2,3,12,16,32,−19,−4,1,−2,0 },   { −13,−21,−48,−102,40,6,7,1,4,11,12,31,−9,−7,0,−1 },   { 7,−19,−13,−31,−109,35,6,3,27,−8,−2,8,27,−4,−6,1 },   { 52,−6,−7,7,25,−10,−8,2,106,−13,−36,3,10,1,1,1 },   { −2,−40,12,12,22,43,−10,−1,−11,−104,16,28,−4,−6,−1,3 },   { −5,−21,−7,−10,−25,−108,21,6,−6,−45,5,9,0,27,1,−3 },   { 12,3,38,−13,−2,−1,−1,35,6,110,−28,−22,5,−3,−1 },   { −5,−4,−4,−14,−11,−18,−121,19,−6,0,−2,−9,−2,−2,25,2 },   { −1,−7,−8,−32,11,12,10,0,−6,−30,−23,−113,15,19,2,3 },   { −1,−3,−4,−3,−8,−8,−17,−124,0,−1,−3,−2,−11,−2,−2,21 },   { 2,0,4,3,26,−15,−7,−10,10,1,22,4,117,−30,−14,1 },   { 0,1,1,1,9,21,−15,−6,2,10,3,21,25,117,−25,−12 },   { −1,0,1,−3,−1,−9,−20,11,−4,−1,−7,−7,−18,−18,−121,19 },   { 0,0,1,0,2,1,7,19,1,4,3,6,6,15,16,124 },   },   {   { 112,−33,−27,−1,−1,−1,−2,0,−42,9,11,1,1,0,0,0 },   { −20,−108,47,26,3,5,2,0,6,34,−11,−10,−2,−1,−1,−1 },   { −30,−28,−104,30,41,6,−1,2,17,12,27,−7,−11,−3,−1,0 },   { −34,−27,−12,−97,−2,24,14,2,−51,8,27,26,3,−7,−2,−2 },   { 29,−22,−20,−51,−26,17,6,3,98,−11,−32,16,4,−4,−4,0 },   { 16,5,30,−28,114,−7,−21,−4,14,−7,−16,8,−21,−3,5,1 },   { 4,32,−6,−23,−2,−7,1,2,5,111,−23,−43,−4,0,3,−1 },   { −10,−19,−17,−23,7,−116,−1,20,−6,−12,−20,0,7,23,6,−3 },   { −12,−4,−29,9,7,27,−9,−4,−40,−10,−106,12,37,−3,4,3 },   { 6,6,4,10,23,−3,123,5,1,−1,−9,1,6,5,−20,−6 },   { 1,11,3,25,0,−19,−5,2,7,45,7,108,13,−33,−10,−3 },   { 1,3,4,6,1,20,−3,123,−4,−1,−5,5,−23,3,11,−12 },   {−3,−1,−7,3,−22,−4,14,−23,−13,−4,−31,16,−112,−18,28,10 },   { 0,−2,1,−4,0,−19,3,14,−2,−14,0,−33,11,−115,−15,29 },   { 1,0,2,2,4,2,19,−1,7,2,12,4,31,−7,119,25 },   { 1,0,−1,0,−1,−3,−1,−12,0,−6,−2,−9,2,−31,23,−121 },   }  } };

Each of Table 15 to Table 17 shows an example in which LFNST kernels areconfigured of 1 LFNST set, wherein one LFNST set is configured of twoLFNST kernel candidates, i.e., an LFNST kernel matrix. The LFNST kernelsthat are proposed in Table 15 to Table 17 are defined according to thegrammar of C/C++ programming languages. And, ing_lfnst_2×8_8×2[1][2][16][16], which is an array that stores LFNSTkernel data, [1] indicates that the kernels are configured of 1 LFNSTset, [2] indicates that each LFNST set is configured of 2 LFNST kernelcandidates, and [16][16] indicates that each LFNST kernel is configuredof a 16×16 matrix. Each 16×16 matrix shown in Table 15 to Table 17represents a matrix that is used in forward LFNST transform. That is,one row becomes one transform basis vector (1×16 vector), which is thenmultiplied by input data that is configured of primary transformcoefficients.

TABLE 15 const int8_t g_lfnst_2×8_8×2[1][2][16][16] = {  { //0   {   { 108,−42,−34,13,−1,−1,−1,0,−37,13,10,−3,1,0,0,1 },   { −25,−99,64,24,−6,3,1,1,0,33,−13,−9,1,0,−1,−1 },   { −35,−28,−83,75,16,−4,−1,2,34,−1,12,−14,−4,0,−3,2 },   { −39,−28,−36,−61,62,4,3,−1,−55,27,35,5,−11,−5,3,−3 },   { 24,−32,−15,−51,41,6,−7,5,83,−12,−45,30,−7,−7,1,1 },   { −16,−29,−34,−38,−83,67,10,−2,5,−18,20,12,11,−7,−6,1 },   { 3,25,−20,−20,−21,1,17,−1,22,101,−30,−49,13,−3,0,−1 },   { −12,−21,−22,−24,−37,−93,56,7,−10,−18,−20,15,15,16,−6,−4 },   { 16,−4,21,−19,−4,−22,20,0,52,−1,91,−38,−40,21,−5,−1 },   { 8,10,11,22,27,36,100,−50,2,5,0,27,−3,−12,−16,3 },   { 1,−11,−1,−17,25,17,14,−15,−7,−52,−23,−92,44,25,−15,7 },   { −4,−2,−11,−3,−15,7,−2,−35,−18,−10,−44,−19,−97,42,28,−7 },   { −2,−6,−3,−8,−11,−29,−43,−109,9,5,12,6,22,−20,0,19 },   { 1,−4,1,−3,−6,−11,17,11,−3,−20,−5,−35,−23,−102,53,18 },   { −2,1,0,−2,−4,−6,−8,13,−9,1,−11,−5,−37,−20,−88,79 },   { −1,0,1,1,3,7,11,9,3,7,7,13,19,42,66,96 },   },   {   { 99,4,−76,−3,17,−2,−4,2,−18,0,12,0,2,−1,0,2 },   { −9,−87,−13,84,26,−21,−6,2,−3,12,3,−10,−1,2,−2,−1 },   { −63,34,−73,−3,68,0,−16,3,29,−9,−4,3,−5,−3,1,1 },   { 28,−3,−1,8,−13,−8,3,6,90,3,−81,−1,26,−3,−6,3 },   { 0,74,−1,70,−31,−67,11,10,−4,−10,11,−9,0,8,1,− 1},   { −33,−19,−65,−2,−75,25,63,−7,6,9,9,−14,2,2,−5,0 },   { 4,11,7,−14,14,−4,−2,2,15,80,9,−87,−23,31,5,−6 },   { 11,34,9,60,1,89,−17,−56,9,3,5,−6,−2,−7,0,5 },   { 12,−10,9,−1,−11,−3,−2,10,70,−12,57,22,−79,−15,26,4 },   { 13,8,25,4,59,−2,106,−19,4,5,2,13,−3,−6,−5,−2 },   { −6,11,−5,5,−8,−7,−7,9,−12,85,−6,47,2,−75,4,25 },   { 4,12,8,22,8,52,13,111,−1,−2,9,−2,12,4,−9,−4 },   { −4,3,−6,8,−2,10,10,11,−41,−12,−73,−6,−67,−1,66,3 },   { −1,4,−5,6,−5,5,−5,−4,2,41,−5,65,−6,62,7,−80 },   { 0,3,−1,1,−3,0,−4,2,−13,−3,−28,0,−65,−10,−105,−11 },   { −2,1,−1,1,0,2,1,1,−1,17,−3,37,−9,74,−11,95 },   }  } },

TABLE 16 const int8_t g_lfnst_2×8_8×2[1][2][16][16] = {  { //0  {110,−34,−36,9,0,−1,−1,0,−37,11,11,−2,1,0,0,1 },   { 20,104,−54,−30,4,−1,−1,−1,1,−33,10,10,0,0,1,1 },   { 35,25,93,−64,−24,4,1,−2,−29,0,−16,12,5,1,3,−1 },   { −22,−39,−33,−91,61,16,1,1,−12,16,10,21,−11,−9,2,−2 },   { 41,−7,10,−7,33,−11,−13,6,97,−10,−58,15,−4,−2,2,3 },   { −2,−27,−33,−34,−91,55,17,−1,35,−14,−2,10,13,−5,−7,2 },   { 2,29,−14,−16,−13,−6,16,−1,16,106,−18,−52,6,0,0,−2 },   { −10,−18,−20,−25,−30,−103,44,14,−7,−15,−20,11,12,18,−4,−4 },   { 16,−3,23,−13,−4,−19,18,3,50,−1,96,−21,−47,16,−2,−3 },   { 7,8,10,21,28,27,109,−37,1,2,−6,25,3,−9,−17,2 },   { −2,11,0,20,−20,−13,−14,16,3,51,16,100,−25,−34,9,5 },   { −4,−4,−10,−5,−18,−14,−19,−101,−6,−1,−23,0,−63,11,20,6 },   { −2,−3,−5,6,−5,23,26,63,−19,−7,−38,−10,−85,30,28,−17 },   { 1,−3,2,−4,−5,−14,14,10,−2,−18,−3,−35,−18,−109,36,31 },   { 2,−2,0,2,3,4,10,−12,9,−2,12,2,36,8,102,−64 },   { 1,0,0,−1,−2,−5,−8,−8,−2,−8,−7,−13,−16,−40,−56,−104 },   },   {   { 102,1,−73,0,15,−3,−4,2,−18,1,11,0,−2,−1,0,2 },   { 9,93,7,−81,−20,16,4,−1,5,−13,−3,9,0,−1,3,1 },   { 61,−32,77,−7,−67,7,13,−4,−27,10,1,−3,5,3,−1,−1 },   { −26,4,0,−9,14,7,−3,−6,−92,0,79,−3,−25,5,6,−3 },   { −2,−68,−2,−77,38,61,−14,−6,6,10,−12,10,−2,−8,0,1 },   { −32,−19,−64,−5,−78,32,57,−9,7,10,10,−16,1,2,−4,0 },   { −4,−12,−8,14,−17,10,−1,−4,−15,−85,−4,84,17,−27,−2,5 },   { 12,32,12,55,4,93,−26,−46,16,5,11,−8,−8,−5,1,4 },   { −11,12,−9,5,12,11,4,−17,−68,15,−60,−14,78,6,−24,−1 },   { 13,7,22,7,26,1,106,−31,6,6,4,19,−6,−9,−2,−1 },   { 6,−11,6,−7,12,6,10,−10,9,−82,4,−55,5,73,−10,−22 },   { 4,11,9,18,12,46,23,112,−1,−1,11,−1,18,3,−12,−1 },   { −3,4,−5,10,−2,14,9,15,−38,−10,−71,−4,−69,9,64,−8 },   { 1,−3,6,−4,5,−4,6,3,−5,−38,−2,−63,−1,−67,3,80 },   { 1,−3,1,−1,2,0,3,−2,13,−3,27,−11,63,−13,103,−24 },   { −2,−1,−1,0,1,1,1,−1,3,17,5,33,12,73,26,94 },   }  } };

TABLE 17 const int8_t g_lfnst_2×8_8×2[1][2][16][16] = {  { //0   {   {− 115,14,34,0,0,1,1,1,42,−4,−11,0,−1,0,0,−1 },   { 11,115,−18,−33,−5,2,−2,−1,2,−37,3,11,1,0,2,1 },   { 27,8,115,−26,−21,−5,−4,−2,−26,1,−22,5,5,3,3,−2 },   { 15,31,15,109,−27,−35,−4,0,15,−9,−9,−23,6,8,−1,2 },   { 13,15,15,6,112,−35,−23,0,16,17,−22,−1,−16,0,6,1 },   { −41,7,2,22,25,−9,−1,−4,−106,−19,42,−2,−2,4,1,−2 },   { 5,−37,2,7,12,−5,−11,1,10,−109,−20,46,5,0,−1,2 },   { 11,8,23,33,24,106,−24,−17,16,−7,33,−4,−16,−16,0,3 } ,   { −11,6,−23,8,4,41,−11,−6,−38.5,−101,−22,39,−4,2,3 } ,   { −6,−6,−10,−14,−27,−16,−120,2,−2.4,8,−11,−4,2,2v0,2 },   { −4,11,−4,24,−7,−2,−11,−7,−8,45,−5,108,19,−36,−4,−1 },   { 1,4,3,6,5,17,−3,125,−3,2,1,5,3,−2,3,−14 },   { −3,1,−8,7,−20,3,10,2,−15,1,−38,12,−113,−8,35,0 },   { 0,2,−3,5,2,18,−3,−7,1,13,−3,32,−2,116,6,−35 },   { 1,−2,1,2,4,2,19,−2,6,−1,12,2,33,0,120,20 },   { 2,−1,−1,0,0,−4,1,−14,3,−7,0,−11.6,−33,18,−121 },   },   {   { 112,−9,−54,−0,5,−3,−4,2,−27,4,10,0,0,0,0,2 },   { −2,−112,24,51,6,4,−2,−1,−13,21,1,−6,0,0,−3,−1 },   { −41,−3,−87,32,72,−12,−12,3,24,−4,5,0,−6,−4,0,2 },   { −30,17,−2,12,8,−4,1,−5,−107,−11,57,−1,−5,3,2,−3 }.   { −2,−29,14,−93,64,34,−12,−4,−11,27,4,10,−13,−6,5,−2 },   { −30,−32,−66,−37,−73,39,31,−2,5,11,24,−10,−1,0,−3,1 },   { −2,−30,−7,−16,−3,9,−8,2,−12,−104,−26,55,18,−5,−3,3 },   { 15,15,19,33,7,82,−24,−18,36,−24,62,6,−32,−4,0,3 },   { −6,21,−11,26,5,74,−9,−23,−35,23,−70,−9,46,−11,−6,3 },   { 10,6,10,12,32,9,117,−21,5,−1,3,25,1,−8,−9,−2 },   { −4,7,−4,11,−17,−9,−19,6,2,51,16,99,13,−51,−2,5 },   { 1,5,5,9,7,30,17,122,−7,1,−5,−1,−3,0,7,0 },   { −3,2,−7,9,−13,3,5,−10,−19,0,−49,10,−102,−19,48,8 },   { −1,3,−7,7,−3,10,−4,−2,−1,18,−9,46,−10,97,5,−64 },   { 0,3,−1,−1,−3,−1,−7,5,−9,−1,−16,−2,−45,−14,−115,−23 },   { 2,−1,0,−1,0,−2,0,0,4,−11,4,−20,10,−59,25,−108 },   }  } };

The above-described embodiment proposes an LFNST kernel that isapplicable to a 2×N or N×2 (N≥8) partition block in the ISP mode (theLFNST kernel corresponding to a top-left 2×8 or 8×2 region in the 2×N orN×2 (N≥8) partition block may be applied) and various applicationmethods of the corresponding LFNST kernel.

Meanwhile, even if an ISP mode is not applied, a 2×N or N×2 coding unitor transform unit may be present (or may exist), and the LFNST kernelthat is applied to the ISP mode may also be applied to such 2×N or N×2coding unit or transform unit (N≥8).

The LFNST kernel that is applied to the 2×N or N×2 coding unit ortransform unit (N≥8) may be an LFNST kernel that is applied to a 2×N orN×2 (N≥8) ISP partition block, and LFNST may be applied by using thesame method as the ISP mode (e.g., application of a corresponding LFNSTkernel to a top-left 2×8 or 8×2 region, embodiments that are describedwith reference to FIG. 19 to FIG. 21, a method for signaling an LFNSTindex in case of an ISP mode, embodiments that are described withreference to Table 6 to Table 17, and so on, may be applied).

Meanwhile, according to the embodiment, in case of the ISP mode, onlyone LFNST candidate may be applied.

Configurations may be made to that, when ISP is not applied, a casewhere the LFNST index value is equal to 2 is removed (or eliminated),and when the LFNST index value is equal to 1, one of two LFNSTcandidates may be selected in accordance with the prediction mode. Inthis case also, when the LFNST index value is equal to 0, this indicatesa case where the LFNST is not applied. Table 2 may be applied by usingthe intra prediction mode, an LFNST set, and a mapping table, andconfigurations may be made so that one or two candidates being includedin an LFNST set that is selected from Table 2 can be selected from theintra prediction mode. The following two embodiments shows methods forselecting an LFNST candidate from an intra prediction mode value.

1. When an intra prediction mode prior to considering a WAIP is referredto as predModeIntra, when a predModeIntra value ranges from 0 to 34,when the predModeIntra value is divided by 2 and the remainder is 0, afirst LFNST candidate may be selected, and when the predModeIntra valueis divided by 2 and the remainder is 1, a second LFNST candidate may beselected.

When a predModeIntra value ranges from 35 to 83, when the predModeIntravalue is divided by 2 and the remainder is 0, a second LFNST candidatemay be selected, and when the predModeIntra value is divided by 2 andthe remainder is 1, a first LFNST candidate may be selected.

2. When an intra prediction mode value that is changed after consideringthe WAIP is referred to as a predModeIntra (wherein the predModeIntramay be given a value ranging from −14 to 83), when the predModeIntravalue is divided by 2 and the remainder is 0, a first LFNST candidatemay be selected, and when the predModeIntra value is divided by 2 andthe remainder is 1, a second LFNST candidate may be selected.

In both of the above-described embodiments, a remainder of dividing thepredModeIntra value by 2, even when the predModeIntra value is anegative number, may be defined. (Herein, if the remainder value (R) isgiven a value of 0 or 1, the remainder may be obtained. That is, thevalue R may be calculated by adjusting a Q value (wherein Q may be givenas a negative number) so that the predModeIntra value can be equal topredModeIntra=2×Q+R.) Based on the VVC standard specification, a firstLFNST candidate may correspond to a case where an LFNST index value(lfnstIdx) is equal to 1, and a second LFNST candidate may correspond toa case where an LFNST index value (lfnstIdx) is equal to 2.

As it is proposed in the above-described embodiments, even in a casewhere the ISP mode is applied and LFNST is applied, a method ofselecting one of two LFNST candidates from an intra prediction modevalue may be applied. Additionally, the corresponding method may beapplied only in a case where a horizontal length and a vertical lengthof a corresponding ISP partition block are both equal to 4 or more.

The following drawings are provided to describe specific examples of thepresent disclosure. Since the specific designations of devices or thedesignations of specific signals/messages/fields illustrated in thedrawings are provided for illustration, technical features of thepresent disclosure are not limited to specific designations used in thefollowing drawings.

FIG. 22 is a flowchart illustrating an operation of a video decodingapparatus according to an embodiment of the present specification.

Each step disclosed in FIG. 22 is based on part of the contentsdescribed above in FIG. 4 to FIG. 22. Accordingly, detailed descriptionsoverlapping with those described above in FIG. 3 to FIG. 21 will beomitted or simplified.

The decoding apparatus 300 may receive residual information from abitstream.

More specifically, the decoding apparatus 300 may decode information onquantized transform coefficients for the target block from the bitstreamand may derive the quantized transform coefficients for the currentblock based on the information on the quantized transform coefficientsfor the current block. The information on the quantized transformcoefficients for the target block may be included in a sequenceparameter set (SPS) or a slice header and may include at least one ofinformation on whether a reduced transform (RST) is applied, informationon the simplification factor, information on a minimum transform size inwhich the reduced transform is applied, information on a maximumtransform size in which the reduced transform is applied, a reducedinverse transform size, and information on a transform index indicatingany one of transform kernel matrices included in a transform set.

Additionally, the decoding apparatus may further receive information onan intra prediction mode for a current block and information on whetheror not ISP is applied to the current block. By receiving and parsingflag information indicating whether or not an ISP code or ISP mode is tobe applied, the decoding apparatus may derive whether or not the currentblock is divided (or split or partitioned) into a predetermined numberof sub-partition transform blocks. Herein, the current block may be acoding block. Moreover, the decoding apparatus may derive sizes andnumber of divided sub-partition blocks through flag informationindicating along which direction the current block is to be divided (orpartitioned).

According to the embodiment, the sub-partition block may be divided (orpartitioned) so that one segment has a length that is less than 4. Forexample, a sub-partition block may be an N×2 (2×N) or N×1 (1×N) block.

The same intra prediction mode may be applied to the sub-partitiontransform block divided from the current block, and the decodingapparatus may derive a prediction sample for each sub-partitiontransform block. That is, the decoding apparatus sequentially performsintra prediction, for example, horizontally or vertically, from left toright, or from top to bottom according to a division form of thesub-partition transform blocks. For the leftmost or uppermost subblock,a reconstructed pixel of an already coded coding block is referred to,as in a conventional intra prediction method. Further, for each side ofthe subsequent internal sub-partition transform block, when it is notadjacent to the previous sub-partition transform block, in order toderive reference pixels adjacent to the corresponding side, areconstructed pixel of an already coded adjacent coding block isreferred to, as in a conventional intra prediction method.

Meanwhile, when the width or height of a sub-partition block is lessthan 4, i.e., when the width or height of a sub-partition block is equalto 2 or 1, an intra prediction mode may be derived by using the symmetrybetween the M×2 (M×1) block and the 2×M (1×M) block.

The decoding apparatus 300 may derive residual information for thecurrent block, i.e., derive transform coefficients by performingdequantization on quantized transform coefficients (S2210).

The derived transform coefficients may be arranged (or aligned) inaccordance with a reversed diagonal scanning order in 4×4 block units,and transform coefficients inside a 4×4 block may also be arranged inaccordance with a reversed diagonal scanning order. That is, transformcoefficients that are processed with dequantization may be arranged inaccordance with a reversed scanning order that is applied in videocodec, such as VVC or HEVC.

The transform coefficients that are derived based on such residualinformation may be dequantized transform coefficients, as describedabove, or quantized transform coefficients. That is, regardless of thequantization or dequantization of the transform coefficients, any datathat may be checked as a non-zero data may be used as the transformcoefficients.

According to the embodiment, the decoding apparatus may derive transformcoefficients that are modified by applying LFNST to the transformcoefficients.

Unlike a primary transform that separately transforms coefficients beingthe transform targets along a vertical or horizontal direction, LFNST isa non-separable transform that applies transform without separating thecoefficients along a specific direction. Such non-separable transmit maybe a low-frequency non-separable transform that applies forwardtransform only in a low-frequency region and not the entire blockregion.

LFNST index information is received as syntax information, and syntaxinformation may be received as a binarized bin string including 0 and 1.

A syntax element of the LFNST index according to the present embodimentmay indicate whether or not inverse LFNST or inverse non-separabletransform is being applied and any one of transform kernel matricesbeing included in a transform set. And, when the transform set includestwo transform kernel matrices, 3 different syntax element values mayexist in the transform index.

That is, according to the embodiment, syntax element values for an LFNSTindex may include 0, which indicates a case where inverse LFNST is notapplied to a target block, 1, which indicates a first transform kernelmatrix of the two transform kernel matrices, and 2, which indicates asecond transform kernel matrix of the two transform kernel matrices.

The intra prediction mode information and LFNST index information may besignaled at a coding unit level.

Additionally, according to the embodiment, the decoding apparatus mayfurther determine whether the transform coefficients are present in asecond region excluding the top-left first region of the current block.And, if the transform coefficients are not present in the second region,the decoding apparatus may parse the LFNST index.

Meanwhile, according to the embodiment, when ISP is applied to thecurrent block, and when the width or height of the sub-partition blockis equal to 2 or less, the encoding apparatus may derive modifiedtransform coefficients that are modified by applying LFNST to thetransform coefficients (S2220).

When the sub-partition block is an N×2 block or a 2×N block, thedecoding apparatus may derive (M×2) or (2×M) number of modifiedtransform coefficients by applying the LFNST matrix to a predeterminednumber of transform coefficients in a top-left M×2 region or 2×M region(M≤N) of the sub-partition block, and, when the sub-partition block isan N×1 block or a 1×N block, the decoding apparatus may derive (M×1) or(1×M) number of modified transform coefficients by applying the LFNSTmatrix to a predetermined number of transform coefficients in a top-leftM×1 region or 1×M region (M≤N) of the sub-partition block.

At this point, the predetermined number may be equal to 8 or 16.

For example, for a top-left M×2 region of a sub-partition block, if M isequal to 8, and if the predetermined number is equal to 16, afterselecting a predetermined LFNST set and an LFNST kernel (e.g., a 16×16matrix) that is derived from a parsed LFNST index, the decodingapparatus may derive an output vector by multiplying the LFNST kernel bya corresponding input vector.

At this point, the (M×2) number of modified transform coefficients andthe (M×1) number of modified transform coefficients may be arrangedaccording to a row-first order in the M×2 region and the M×1 region,respectively.

Additionally, for example, for a top-left 2×M region of a sub-partitionblock, if M is equal to 8, and if the predetermined number is equal to8, after selecting a predetermined LFNST set and an LFNST kernel (e.g.,a 16×8 matrix) that is derived from a parsed LFNST index, the decodingapparatus may derive an output vector by multiplying the LFNST kernel bya corresponding input vector.

At this point, the (2×M) number of modified transform coefficients andthe (1×M) number of modified transform coefficients may be arrangedaccording to a column-first order in the 2×M region and the 1×M region,respectively.

A remaining sub-partition block region in which the modified transformcoefficients are not arranged may be filled with zeros (0s). This meansthat zero-out is performed for the LFNST.

Meanwhile, an LFNST set for deriving an LFNST matrix may be derivedbased on an intra prediction mode of the current block, and a number ofthe LFNST sets may be equal to any one of 7, 4, 3, 2, and 1.

At this point, since it may be regarded that mode number 2 (Mode 2) inthe M×2 (M×1) block is symmetric with mode number 66 (Mode 66) in the2×M (1×M) block, the same LFNST set may be applied to the 2×M (1×M)block and the M×2 (M×1) block. In order to apply the LFNST set appliedto the M×2 (M×1) block to the 2×M (1×M) block, the LFNST set is selectedbased on Mode 2 instead of Mode 66. That is, after transposing inputdata of the 2×M (1×M) block before applying the LFNST, the LFNST maythen be applied.

According to the embodiment, LFNST for a sub-partition block having awidth or height that is equal to 2 or less may be applied to the currentblock, even if ISP is not applied to the current block. That is, whenthe width or height of the current block (coding block or transformblock) is equal to 2 or less, the above-described LFNST set selection,size of the LFNST matrix, array (or arrangement) of the modifiedtransform coefficients, and so on, may be applied.

The decoding apparatus may derive residual samples for the current blockbased on inverse primary transform on the modified transformcoefficients (S2230).

At this point, a general separable transform may be used as the inverseprimary transform, and the above-described MTS may also be used.

Subsequently, the decoding apparatus 200 may generate reconstructedsamples based on residual samples for the current block and predictionsamples for the current block (S2240).

The following drawings are provided to describe specific examples of thepresent disclosure. Since the specific designations of devices or thedesignations of specific signals/messages/fields illustrated in thedrawings are provided for illustration, technical features of thepresent disclosure are not limited to specific designations used in thefollowing drawings.

FIG. 23 is a flowchart illustrating an operation of a video encodingapparatus according to an embodiment of the present specification.

Each step disclosed in FIG. 23 is based on some of the contentsdescribed above in FIG. 4 to FIG. 21. Accordingly, detailed descriptionsoverlapping with those described above in FIG. 2 and FIG. 4 to FIG. 21will be omitted or simplified.

The encoding apparatus according to an embodiment may first deriveprediction samples for the current block based on the intra predictionmode applied to the current block and may derive residual samples forthe current block based on the prediction samples (S2310).

The encoding apparatus may perform prediction per sub-partitiontransform block, when ISP is being applied to the current block.

The encoding apparatus may determine whether to apply ISP coding or anISP mode to the current block, i.e., the coding block, may determine adirection in which the current block will be divided according to thedetermination result, and may derive the size and number of dividedsub-blocks.

According to the embodiment, the sub-partition block may be divided (orpartitioned) so that one segment has a length that is less than 4. Forexample, a sub-partition block may be an N×2 (2×N) or N×1 (1×N) block.

The same intra prediction mode may be applied to the sub-partitiontransform block divided from the current block, and the encodingapparatus may derive a prediction sample for each sub-partitiontransform block. That is, the encoding apparatus sequentially performsintra prediction, for example, horizontally or vertically, from left toright, or from top to bottom according to a division form of thesub-partition transform blocks. For the leftmost or uppermost subblock,a reconstructed pixel of an already coded coding block is referred to,as in a conventional intra prediction method. Further, for each side ofthe subsequent internal sub-partition transform block, when it is notadjacent to the previous sub-partition transform block, in order toderive reference pixels adjacent to the corresponding side, areconstructed pixel of an already coded adjacent coding block isreferred to, as in a conventional intra prediction method.

Meanwhile, when the width or height of a sub-partition block is lessthan 4, i.e., when the width or height of a sub-partition block is equalto 2 or 1, an intra prediction mode may be derived by using the symmetrybetween the M×2 (M×1) block and the 2×M (1×M) block.

Further, the encoding apparatus may derive transform coefficients forthe current block based on a primary transform of the residual samples(S2320).

The primary transform may be performed through a plurality of transformkernels, and, in this case, a transform kernel may be selected based onthe intra prediction mode.

The encoding apparatus may determine whether or not to perform asecondary transform, or non-separable transform, more specifically,LFNST, for the transform coefficients of the current block, and mayderive modified transform coefficients by applying LFNST to thetransform coefficients. And, when ISP is applied to the current block,and when the width or height of a sub-partition block is equal to 2 orless, the encoding apparatus may apply LFNST to the transformcoefficients and may derive modified transform coefficients that aremodified by applying LFNST to the transform coefficients (S2330).

Unlike a primary transform that separately transforms coefficients beingthe transform targets along a vertical or horizontal direction, LFNST isa non-separable transform that applies transform without separating thecoefficients along a specific direction. Such non-separable transmit maybe a low-frequency non-separable transform that applies transform onlyin a low-frequency region and not the entire target block, which is thetransform target.

When the sub-partition block is an N×1 block or a 1×N block, theencoding apparatus may derive 8 or 16 modified transform coefficients byapplying the LFNST matrix to transform coefficients in a top-left M×1region or 1×M region (M≤N) of the sub-partition block.

For example, when a size of the sub-partition block is equal to N×2(2×N), LFNST may be applied to a top-left M×2 (2×M) region (whereinM≤N). For example, when M=8, since the corresponding top-left regionbecomes 8×2 (2×8), a region having 16 residual signals may be the inputof a forward LFNST, and an R×16 (R≤16) forward transform matrix may beapplied. Herein, R may be equal to 8 or 16.

Additionally, when a size of the sub-partition block is equal to N×1(1×N), LFNST may be applied to a top-left M×1 (1×M) region (whereinM≤N). For example, when M=16, since the corresponding top-left regionbecomes 16×1 (1×16), a region having 16 residual signals may be theinput of a forward LFNST, and an R×16 (R≤16) forward transform matrixmay be applied. In this case, also, R may be equal to 8 or 16.

At this point, transform coefficients of the M×2 region and transformcoefficients of the M×1 region may be arranged in a one-dimensionalarray according to a row-first order so as to be calculated with theLFNST matrix, and transform coefficients of the 2×M region and transformcoefficients of the 1×M region may be arranged in a one-dimensionalarray according to a column-first order so as to be calculated with theLFNST matrix.

The encoding apparatus may perform zero-out with zeros (0s) on aremaining sub-partition block region in which the modified transformcoefficients are not arranged.

Meanwhile, an LFNST set for deriving an LFNST matrix may be derivedbased on an intra prediction mode of the current block, and a number ofthe LFNST sets may be equal to any one of 7, 4, 3, 2, and 1.

At this point, since it may be regarded that mode number 2 (Mode 2) inthe M×2 (M×1) block is symmetric with mode number 66 (Mode 66) in the2×M (1×M) block, the same LFNST set may be applied to the 2×M (1×M)block and the M×2 (M×1) block. In order to apply the LFNST set appliedto the M×2 (M×1) block to the 2×M (1×M) block, the LFNST set is selectedbased on Mode 2 instead of Mode 66. That is, after transposing inputdata of the 2×M (1×M) block before applying the LFNST, the LFNST maythen be applied.

According to the embodiment, LFNST for a sub-partition block having awidth or height that is equal to 2 or less may be applied to the currentblock, even if ISP is not applied to the current block. That is, whenthe width or height of the current block (coding block or transformblock) is equal to 2 or less, the above-described LFNST set selection,size of the LFNST matrix, array (or arrangement) of the modifiedtransform coefficients, and so on, may be applied.

The encoding apparatus may derive quantized transform coefficients byperforming quantization based on the modified transform coefficients ofthe current block, and the encoding apparatus may, then, encode andoutput information on the quantized transform coefficients, and, in caseLFNST is applicable (i.e., in case LFNST can be applied), imageinformation including LFNST index information indicating an LFNST matrix(S2340).

The encoding apparatus may generate residual information includinginformation on quantized transform coefficients. The residualinformation may include the above-described transform relatedinformation/syntax element. The encoding apparatus may encodeimage/video information including residual information and, then, outputthe encoded image/video information in the form of a bitstream.

More specifically, the encoding apparatus may generate information onthe quantized transform coefficients and encode the information on thegenerated quantized transform coefficients.

The syntax element of the LFNST index according to the presentembodiment may indicate whether (inverse) LFNST is applied and any oneof the LFNST matrices included in the LFNST set, and when the LFNST setincludes two transform kernel matrices, three values of the syntaxelement of the LFNST index may exist.

According to an embodiment, when a division tree structure for thecurrent block is a dual tree type, an LFNST index may be encoded foreach of a luma block and a chroma block.

According to an embodiment, the syntax element value for the transformindex may be derived as 0 indicating a case in which (inverse) LFNST isnot applied to the current block, 1 indicating a first LFNST matrixamong LFNST matrices, and 2 indicating a second LFNST matrix among LFNSTmatrices.

1. An image decoding method performed by a decoding apparatus,comprising: deriving transform coefficients based on residualinformation for a current block; deriving modified transformcoefficients by applying LFNST to the transform coefficients based onISP being applied to the current block and a width or height of asub-partition block being equal to 2 or less; deriving residual samplesfor the current block based on inverse primary transform on the modifiedtransform coefficients; and generating a reconstructed picture based onresidual samples for the current block, wherein, based on thesub-partition block being an N×2 block or a 2×N block, (M×2) or (2×M)number of modified transform coefficients are derived by applying anLFNST matrix to a predetermined number of transform coefficients in atop-left M×2 region or 2×M region (M≤N) of the sub-partition block. 2.The image decoding method of claim 1, wherein, based on thesub-partition block being an N×1 block or a 1×N block, (M×1) or (1×M)number of modified transform coefficients are derived by applying theLFNST matrix to a predetermined number of transform coefficients in atop-left M×1 region or 1×M region (M≤N) of the sub-partition block. 3.The image decoding method of claim 2, wherein the predetermined numberis equal to 8 or
 16. 4. The image decoding method of claim 2, whereinthe (M×2) number of modified transform coefficients and the (M×1) numberof modified transform coefficients are arranged according to a row-firstorder in the M×2 region and the M×1 region, respectively.
 5. The imagedecoding method of claim 2, wherein the (2×M) number of modifiedtransform coefficients and the (1×M) number of modified transformcoefficients are arranged according to a column-first order in the 2×Mregion and the 1×M region, respectively.
 6. The image decoding method ofclaim 1, wherein a remaining sub-partition block region in which themodified transform coefficients are not arranged is filled with zeros(0s).
 7. The image decoding method of claim 1, wherein the LFNST matrixis derived from a plurality of LFNST sets based on an intra predictionmode of the current block, and wherein a number of the LFNST sets isequal to any one of 7, 4, 3, 2, and
 1. 8. An image encoding methodperformed by an encoding apparatus, comprising: deriving residualsamples for a current block based on prediction samples for the currentblock; deriving transform coefficients by applying a primary transformof the residual samples; deriving modified transform coefficients thatare modified by applying LFNST to the transform coefficients based onISP being applied to the current block and a width or height of asub-partition block being equal to 2 or less; and encoding residualinformation including information on the modified transformcoefficients, wherein, based on the sub-partition block being an N×2block or a 2×N block, 8 or 16 modified transform coefficients arederived by applying an LFNST matrix to transform coefficients in atop-left M×2 region or 2×M region (M≤N) of the sub-partition block. 9.The image encoding method of claim 8, wherein, based on thesub-partition block being an N×1 block or a 1×N block, 8 or 16 modifiedtransform coefficients are derived by applying the LFNST matrix totransform coefficients in a top-left M×1 region or 1×M region (M≤N) ofthe sub-partition block.
 10. The image encoding method of claim 9,wherein transform coefficients of the M×2 region and transformcoefficients of the M×1 region are arranged in a one-dimensional arrayaccording to a row-first order so as to be calculated with the LFNSTmatrix.
 11. The image encoding method of claim 9, wherein transformcoefficients of the 2×M region and transform coefficients of the 1×Mregion are arranged in a one-dimensional array according to acolumn-first order so as to be calculated with the LFNST matrix.
 12. Theimage encoding method of claim 8, wherein a remaining sub-partitionblock region in which the modified transform coefficients are notarranged is filled with zeros (0s).
 13. The image encoding method ofclaim 8, wherein the LFNST matrix is derived from a plurality of LFNSTsets based on an intra prediction mode of the current block, and whereina number of the LFNST sets is equal to any one of 7, 4, 3, 2, and
 1. 14.A non-transitory computer-readable digital storage medium which stores abitstream generated by a method, deriving residual samples for a currentblock based on prediction samples for the current block; derivingtransform coefficients by applying a primary transform of the residualsamples; deriving modified transform coefficients that are modified byapplying LFNST to the transform coefficients based on ISP being appliedto the current block and a width or height of a sub-partition blockbeing equal to 2 or less; and encoding residual information includinginformation on the modified transform coefficients to generate thebitstream, wherein, based on the sub-partition block being an N×2 blockor a 2×N block, 8 or 16 modified transform coefficients are derived byapplying an LFNST matrix to transform coefficients in a top-left M×2region or 2×M region (M≤N) of the sub-partition block.